One game springs to mind:
The Titans were trailing by 7 going into the 2 minute warning. With 31 seconds left, Marcus Mariota found TE Luke Stocker for a 1 yard TD. The Titans head coach Mike Vrabel elected to go for a two-point try to win the game, that subsequently failed.
I came across some more non-playoff games while trying to find out if it has ever happened in a playoff game (I am 99.9% certain it has not, but I have no citation):
- The first time it happened was in Week 15 of 2002, when the Vikings were successful on a 2-point conversion (despite Daunte Culpepper fumbling the snap) with 5 seconds left against the Saints and won 32-31.
- The Texans were unsuccessful on a 2-point conversion with 14 seconds left against the Titans in Week 17 of 2011 and lost 23-22.
- The Raiders were successful on a 2-point conversion with 47 seconds left against the Saints in Week 1 of 2016 and won 35-34.
- The Vikings were successful on a 2-point conversion with no time left against the 49ers in the preseason of 2017 and won 32-31.
- The Chargers were successful on a 2-point conversion with 4 seconds left against the Chiefs in Week 15 of 2018 and won 29-28.
- The Broncos were successful on a 2-point conversion with 31 seconds left against the Bears in Week 2 of 2019 to take a 14-13 lead, but lost 16-14 after the Bears made a field goal as time expired.
- The Dolphins were unsuccessful on a 2-point conversion with 6 seconds left against the Redskins in Week 6 of 2019 and lost 17-16.