FORT LAUDERDALE, Fla. — On a sunny January day nearly 50 years ago, Nick Buoniconti, Jim Kiick and Jake Scott ran off the field at the Los Angeles Memorial Coliseum not just as Super Bowl champions, but as part of the only team in N.F.L. history to complete a perfect season.

They all played pivotal roles in the 1972 Miami Dolphins’ final victory. Scott, the gifted free safety, was named most valuable player of that game. Buoniconti, the pugnacious leader of the team’s No-Name Defense, made a key interception. Kiick, the versatile running back, scored the winning touchdown.

Within a few years, they all left the N.F.L. and went their separate ways. Save for the occasional team reunion or charity event, they rarely met.

Decades later, the three cornerstones of that team have been united in a way they never expected: They all had the most severe form of chronic traumatic encephalopathy, or C.T.E., the degenerative brain disease associated with repeated head hits. Buoniconti, Kiick and Scott died within 16 months of each other in 2019 and 2020, and for the first time, their families discussed caring for the exceptionally different N.F.L. veterans as they faced the consequences of their drive for football glory.

It is difficult to determine what accounts for this unusual cluster of brain trauma on a roster of about 45 players. Their coach, Don Shula, drove his players hard, even holding four practices a day during training camp. Players from that era lowered their heads when running with the ball or tackling, which could explain the extensive damage that Buoniconti, Kiick and Scott had to their frontal lobes and brain stems. They all played football from a young age and absorbed tens of thousands of head hits.

“There’s something about that team,” said Dr. Ann McKee, who has diagnosed C.T.E., which can only be diagnosed definitively after death, in hundreds of former football players, including Buoniconti, Kiick and Scott, as well as three other 1972 Dolphins, Earl Morrall, Bill Stanfill and Bob Kuechenberg.

Buoniconti, Kiick and Scott shared similar afflictions, McKee said. They began showing symptoms in their late 60s and early 70s and had issues associated with Parkinson’s disease, like tremors, frequent falling and shuffling gaits. Later, their memories failed.

Their demise is a reminder that reaching the pinnacle of the sport can unspool chaotically far from the bright lights of football stadiums. The caregivers — wives, children, siblings — rarely have the physical and emotional training or financial wherewithal to match the immense task of caring for men whose bodies and minds can wither with shocking speed.

“He was the guy I looked up to my whole life, so to see him deteriorate, I tried to shut my eyes to it,” said Austin Kiick, who in his 20s took care of his father. “No kid wants to put their parents in a home. But you could see things were happening to him.”

Jim Kiick

When Jim Kiick retired, his notoriety and laid back style served him well as an investigator for the public defender’s office in South Florida. He had, after all, worked in the shadow of his more famous teammate, Hall of Fame fullback Larry Csonka. But Kiick’s quiet nature camouflaged symptoms that first started to appear in his mid-60s when he was still active and going to the gym every day.

Kiick was all about routines — meticulous at home and devoted to Austin and his sister, Allie. He picked them up from school and drove them to their after-school practices, where he would get worked up watching them play football, tennis and other sports. He also knew how to console them after tough losses.

“It was like he was out there playing,” said Allie, now 26. “If we lost, he lost.”

Though he had a reputation as standoffish, Kiick never turned down a request for an autograph. He loved watching “Extreme Makeover” and would cry when the families were given the keys to their renovated homes.

After he left the public defender’s office, Kiick’s memory slipped. He kept losing his A.T.M. card. He would drive to places and not know how he got there. He sometimes called Austin and Allie 20 and 30 times in a row.

Allie became a pro tennis player and often had to block her dad’s calls to get some sleep when she was on the road. Austin, now 33, returned to Fort Lauderdale to care for his dad but had no medical training.

“I’m doing this on a whim, not understanding what’s going on,” he said.

Kiick used to neatly fold his clothing, but Austin would find it scattered around his home. Austin programmed the tracking function in his dad’s cellphone and discovered that Jim had been going to the gym several times a day. Finding four bags of takeout food in the kitchen, Austin found out that Jim had bought a hamburger from a nearby restaurant, forgot about it, and repeated the trip three more times.

In 2014, Kiick traveled to Alaska with his old teammate, Mercury Morris, to film a documentary, “The Perfect Backfield,” about their time with Csonka on the Dolphins.

Morris checked in for their flight and went to the gate, but couldn’t find Kiick there. Then he got a phone call. “Merc, where are you?,” Kiick asked. “No, Jim, where are you?” Morris responded. Kiick was in his car in the parking lot but couldn’t figure out what to do next. Morris coaxed him out and directed him to the terminal.

“He was drawing blanks on what to do: Not where to go, but what to do,” Morris said. “And that was just the start of it.”

Two years later, Kiick’s ankles and legs swelled from a lack of blood flow. Austin had to give him turn-by-turn directions to get to the hospital. There, the staff recognized Kiick’s dementia and would not release him unless he was sent to an assisted living facility.

“We probably lost our dad five, six years ago, not one, two years ago,” Austin said.

There, Kiick fell into safe routines — breakfast with a cup of orange juice every morning at 7:30, workouts three times a week. Austin said his father was one of the younger residents, and one of the most popular.

Kiick, who was not wealthy, was approved for the N.F.L.’s Plan 88, which provides up to $160,000 for medical and custodial care for players with dementia and other diseases. He eventually received money from the league’s class action concussion settlement. But the coronavirus pandemic cut him off from the routines that kept him centered and, his kids said, accelerated his demise before he died in June 2020.

“You could have given me a million dollars and it wouldn’t be enough,” Austin said of the settlement. “I’d give back every dollar to have my dad back.”

Jake Scott

Jake Scott was happy to leave football behind. When he was with the Dolphins, he built a cabin in the Rocky Mountains and after leaving the N.F.L. made a beeline for Hawaii, where he was a regular at Tahiti Nui, a bar where everyone knew his name and no one mentioned the Dolphins.

“He’d have a whole conversation with people who didn’t have a clue who he was,” Scott’s friend, Richie Hill, said.

He would visit Key West to stay with his sister, Rita Scott Fabal, and her husband, Randy Fabal, who built a small home for him. Scott saw friends in Atlanta and Athens, Ga., where he had starred in college.

Even before he became ill, friends and family looked after him, though he didn’t need much. Rita took care of his finances. Randy would book his trips, often organized on a whim. He took road trips, including to Alaska. He earned the gratitude of his neighbors when he rode out Hurricane Iniki with them and had generators and Spam flown in.

Like Kiick, Scott began to show signs of decline in his late 60s. He would call Rita for phone numbers but then forget them. She would send him a list of his medications so he could hand it to doctors. Randy would have to review airline reservations with him numerous times. Scott told Rita that he gave her phone number to friends “in case I fall and die.” Rita bought him a big calendar and “he’d go over it and over it,” she said.

He had a friend write down a list of his injuries: fractured wrists, damaged knees, broken ankles, concussions dating back to Pop Warner. In the N.F.L., Scott was knocked out by Bills fullback Jim Braxton. After the game, Scott roamed into Buffalo’s locker room. O.J. Simpson walked him to the Dolphins’ locker room and said, “I think this one belongs to you.”

Scott sensed he was slipping and in 2017, when he was 72, he took tests to provide a benchmark that could be used later to measure his decline.

Rita and Randy and his friends didn’t call attention to his shaky memory or, alarmingly, the growing number of falls. (He would say jokingly that he “tilted.”) Someone would drive him to the airport, make sure he got on the plane and another friend would pick him up at his destination.

“Everybody let Jake be Jake,” Rita said. “He needed to be independent, or at least appear independent.”

Air travel can be disorienting, but Scott liked planes because, he said, if he fell, someone was there to help. He would give flight attendants $100 and tell them to bring him beers until the money ran out.

In 2020, he flew to Atlanta to visit his friend, Jim Hancock, who noticed that Scott had severe tinnitus and slept with the television on to mute the ringing in his ears.

After returning from lunch one day, Scott fell down the steps to a basement apartment and injured his neck. He responded well to a surgery to repair it, but then his left lung collapsed and he had to be sedated and put on a respirator. A few days later, he died.

Rita and Randy sent Scott’s ashes to Kauai, Hawaii, where his old friends scattered them off the coast. They also threw in a wreath with a big Budweiser can strung to it. They wore T-shirts emblazoned with the words, “It’s a Jake thing. You wouldn’t understand.”

Nick Buoniconti

Lynn Buoniconti started to see her husband slipping in 2011 even as he continued to fool many others. She remembered that, decades after Nick had stopped playing, he was still just as charismatic as when he negotiated player contracts, ran companies, co-hosted HBO’s “Inside the N.F.L.,” and raised hundreds of millions of dollars for spinal cord research.

But in private, Lynn saw a different Nick. She’d watched Dr. McKee testify to Congress about C.T.E. in football players in 2009, and what she said worried Lynn. Nick’s driving had become erratic: He would turn impulsively or stop at a light and not know what to do. At dinner with friends, he would make inappropriate comments without realizing he had.

He began to fall. His left hand had a tremor. He became anxious giving speeches and other activities that he had never worried about before. His concentration frayed. When Lynn told him that she had been diagnosed with breast cancer, in 2016, he responded by asking when dinner would be ready.

As Nick deteriorated, so did the confidence and pride that made him so successful as a player, and he agreed with Lynn that they should seek answers. In December 2015, doctors at the Feinstein Institutes for Medical Research on Long Island found that Nick’s brain was shrinking, a clear sign that the symptoms weren’t just related to aging.

“Finally, someone validated that this wasn’t just that he was older,” Lynn said. “We knew from that diagnosis it was really three to five years.”

Lynn had to hire round-the-clock home care, but many workers quit because of Nick’s outbursts. He was approved for the N.F.L.’s Plan 88, but even so, Lynn said they spent about $150,000 a year out of pocket to care for Nick, who could no longer put on a shirt by himself.

Forever tenacious, Nick felt that the N.F.L. was not doing enough for former players. So in 2017, he and Lynn went to Boston so Dr. McKee could see him as he declined, not just study his brain after he was gone. He insisted on holding a news conference where he pledged his brain to science: He wanted people to see that if he was admitting that he was sick, others could do the same.

“Nick called the shots until he couldn’t,” Lynn said.

In his last days, Nick slipped into a coma. Looking for anything to raise his spirits, Lynn invited close friends to see him. He did not visibly respond.

Then Mary Anne Shula, Don Shula’s wife, called to check on Lynn and Nick. During their days together, Shula and Nick locked horns, though they later joked about their stubbornness.

Shula came on the line and in his playfully gruff way, said, “Nick, you’re still a pain in the [expletive], but I love you.” Lynn saw a small tear well up on the side of Nick’s eye.

He died three days later.