“Before the system was in place, they would give you a write-up right off the bat,” he said. “Even if you were late by the train, it just goes in your folder.”

In recent years, Mr. Greer said, the notices have served a dual purpose: absolving guards on particular mornings — typically involving the E and No. 6 trains, in his case — and proving to supervisors that employees probably should have been taken at their word all along.

“You have the proof,” he said. “That means they can trust you.”

Tom Weppler, 48, an information technology engineer from City Island, first asked for a delay verification in October 2012, hoping to piece together a morning in which he reported to work an hour late. (Because he had to leave early that afternoon, he recalled, he had “begged” his boss days earlier to let him start at 8:30 a.m. instead of 9:30 a.m.)

In fact, Mr. Weppler was told one day later, train trouble had touched 16 subway lines. “As a result,” the authority’s response read, “any one delay lasted up to 395 minutes,” or 6 hours 35 minutes.

“I could have gone to see a really long movie and had lunch if I’d known I was going to be excused for that long,” he said.

Some riders wondered if the generous delay estimates had compromised the notification system’s credibility, though shorter delays have also posed a potential problem.

The authority said it considered any train that reached its terminal destination “no more than five minutes late” to be on time, suggesting that any smaller holdups might go unrecorded. A spokesman added that the agency could not recall receiving a request to confirm a delay of less than five minutes.