According to the immigration agency, its employees do not pick up the mail from the lockbox. The United States Department of the Treasury manages the process but uses a courier service that picks up the mail from post offices each morning. Express Mail items, the agency said, are picked up in the afternoon. Also, items are delivered to the lockbox by the courier services FedEx, DHL and the United Parcel Service.

In a memo after DACA was rescinded, the Department of Homeland Security said renewals had to be “accepted” by Oct. 5. Immigration lawyers contrasted that with a permanent residency opportunity the government offered in 2001, when applications only had to be postmarked by the deadline of April 30. Acknowledging the high volume of applications, the government offered a grace period for paperwork to be received by May 3.

Immigration lawyers questioned why that was not an option in this case.

Hasan Shafiqullah, director of the immigration unit of the Legal Aid Society in New York, was disturbed by the lack of compassion from the agency.

“From the clients’ perspective, they did the right thing,” Mr. Shafiqullah said. “Filing three weeks before should be sufficient, and U.S.C.I.S. needs to recognize that and needs to exercise discretion.”

According to an Oct. 18 deposition of an immigration official conducted as part of a federal lawsuit in Brooklyn, 4,000 DACA applications arrived late. One hundred and fifty-four thousand people were eligible to apply for renewal and 132,000 applications were received on time.

Tata Camara, 32, who came to the United States at age 15 from Guinea, sent her application on Sept. 29; she said it was the earliest she could afford to apply, since that day she got a donation for the $495 application fee from the New Economy Project, a New York nonprofit. BronxWorks, a legal services agency, helped her prepare her application.