WASHINGTON — With the Aug. 31 deadline for a U.S. withdrawal from Afghanistan rapidly approaching, the Pentagon has sharply ramped up the speed of evacuations from the Kabul airport, flying out 21,600 people in 24 hours, Defense Department officials said Tuesday. But bottlenecks in the system, and President Biden’s insistence that all troops leave the country by the end of the month, may prevent the military from keeping that pace.

The race against time means that the 5,800 Marines and soldiers at Hamid Karzai International Airport must try to evacuate thousands more Americans and Afghan allies, and then get themselves out, somehow erasing the detritus of 20 years of war in Afghanistan in the next seven days.

That process began on Tuesday, as John F. Kirby, the Pentagon spokesman, said that several hundred headquarters, maintenance and other support troops who were not essential to the escalating evacuation operation had left the country.

Defense officials have been loath to say publicly, though, what seems increasingly clear: Some people will be left behind.

Since Aug. 14, when Kabul fell to the Taliban, more than 70,700 people have been evacuated from Afghanistan as of Tuesday evening, Mr. Biden said.

That is well below the number of American citizens, foreign nationals and Afghan allies who are trying to get out. “We’re trying to get as many out as we can,” said John F. Kirby, the chief Pentagon spokesman. He said the American troops at the Kabul airport want to “continue this pace as aggressively as we can.”

But for all of Mr. Biden’s insistence on sticking to his withdrawal deadline, neither the Department of Homeland Security nor the State Department has been able to ratchet up the vetting and processing times to the levels needed to meet demand.

A U.S. official said it was taking as long as 12 hours for immigration officers at Al Udeid Air Base outside Doha, Qatar, to check arriving Afghans against the National Counterterrorism Center’s watch list. The official said that vetting and screening processes needed to move faster to prevent the evacuee pipeline from clogging up again at Al Udeid, the largest base receiving Afghans, as it did for several hours last week.

The Taliban have warned of “consequences” if the U.S. military stays beyond the deadline. And on Tuesday, a Taliban spokesman said the group’s fighters would physically block Afghans from going to the airport.

The Pentagon has opened military bases in Virginia, Texas, Wisconsin and New Jersey to provide temporary housing for Afghan refugees, and will probably add more in the coming days, officials said.

Mr. Kirby said Afghan allies of the United States, who fear reprisals from the Taliban, are still being processed at the Kabul airport, although the gates of the airport have been shuttered several times over the past week because of the surge of people.

The United States will continue to evacuate Afghans until the last couple of days of its drawdown of troops and equipment, when flights are expected to be filled mostly with military troops and equipment, as well as any Americans desiring to leave. Dozens of Afghan commandos — trained by the United States — are also at the airport and must be evacuated.

For the military, part of the problem is the sheer scale of moving so many people so quickly, with so little advanced notice. For instance, the C-17 military planes, which are transporting 400 people per load, have one or two bathrooms on them, and the flight from Kabul to Qatar is four hours.

Once the flights arrive at Al Udeid in Qatar and other intermediate bases in the Middle East and Europe, evacuees are vetted by Homeland Security and State Department officials, who determine if they qualify to enter the United States.

The military is treating the Taliban’s red line on Aug. 31 seriously in part because, despite tough talk from Taliban spokesmen, some of the group’s commanders have been cooperating with the U.S. military and allowing many people to get to the airport. In addition, the American military and the Taliban have cooperated against the threat of attacks from the Islamic State.

But after Aug. 31, all bets are off, a senior U.S. official said.

With so many people at the Kabul airport, in Doha and at other bases, concerns are rising about sanitation, food and water. The C-17 planes taking refugees out of Afghanistan are turning around and bringing in extra dumpsters, portable hand-washing stations, refrigerated trucks to keep water cool, and food and water.

Over the past four days, Defense Department officials said, three babies were born to evacuees. One woman went into labor on Saturday during a flight landing at Ramstein Air Base in Germany, Air Force officials said. The aircraft commander descended to a lower altitude to increase air pressure in the jet, a decision that officials said saved the mother’s life because she had low blood pressure. When the plane landed, medics rushed aboard and delivered the baby — a girl — in the cargo bay. All three babies are in good condition, Mr. Kirby said Tuesday.

After receiving a classified briefing Monday night, Representative Adam B. Schiff, a California Democrat who heads the House Intelligence Committee, said the Aug. 31 deadline for withdrawing U.S. troops from Kabul was unrealistic.

“I think it’s possible, but I think it’s very unlikely,” Mr. Schiff told reporters. Using the abbreviation for special immigrant visas, he added, “Given the number of Americans who still need to be evacuated, the number of S.I.V.s, the number of others who are members of the Afghan press, civil society leaders, women leaders — it’s hard for me to imagine all of that can be accomplished between now and the end of the month.”