I have written before about a local high school team that doesn’t punt. And last week I went to watch them play. They were in an unusually close matchup. In fact, they were up by only 6 points with 1:42 left in the 4th quarter. They were on their own 22 yard line and facing 4th and 10. They went for it, because, as I said, they don’t punt. They got 9.9 yards on 4th down. Their opponent scored a touchdown in only two plays. But they missed the XPA. Scored tied at 40. (But then the non-punting team drove 85 yards in 90 seconds, and they kicked a chip-shot FG with time expiring.) I don’t think they have lost in a couple of years.

Out of curiosity, I went back a little further, looking at teams who faced the impossible (down 15+ points in the fourth quarter, facing fourth down on their own side of the field).

4 times teams were in that position, and went on to win the game. None of them went for it on fourth down.

Two of them were the same team!
In 2003, the Ravens were down 41-24 in early in the fourth quarter to Seattle. Facing 4th and 12 from their own 20, they punted. Then on their next drive (same score), they punted on 4th and 10 from their own 30. Seattle muffed the punt, with Baltimore recovering. But this wasn’t the key to the comeback, because Jamal Lewis fumbled on the next play. With 7 minutes left, Baltimore forced a Seattle punt, blocked it and returned it for a TD. Baltimore scored on its next drive to cut the score to 41-38. Baltimore kicked a last second field goal to send the game to overtime, and won in overtime.
https://www.pro-football-reference.com/boxscores/200311230rav.htm

In 2006, the Titans were trailing the Giants 21-0 early in the fourth. They punted, and then immediately Pacman Jones intercepted Eli Manning, and eventually turned it into a TD. They forced 2 more punts, and turned them into touchdowns to tie the game. With 32 seconds left, Pacman intercepted Manning again to set up the game-winning FG.

https://www.pro-football-reference.com/boxscores/200611260oti.htm

In 2011, the Broncos trailed the Dolphins 15-0 midway through the 4th. They punted from their own 31. This was a Tim Tebow special. He led the Broncos to 2 touchdowns and a 2-point conversion in the final 3 minutes, thanks to a successful onside kick. The Broncos won in overtime after a strip sack of Matt Moore.

https://www.pro-football-reference.com/boxscores/201110230mia.htm