You must have missed some of the most unbelievable endings to overtime games in the last several years, including one earlier this season between Pittsburgh and Cleveland at Heinz Field. In that game, Pittsburgh lined up to kick the winning field goal in overtime on third down. Cleveland blocked the kick, but Pittsburgh recovered the ball behind the line of scrimmage (if it had gone beyond the line of scrimmage it would have been a "deflected" kick and therefore Cleveland's ball on the missed field goal) and then successfully won the game with a field goal on 4th down.
The Giants played an overtime game against Philadelphia a few years back, and a similar thing happened. The Giants blocked the winning field goal attempt, but Philadelphia recovered the ball in their own backfield and ran it into the end zone for a game-winning touchdown.
A pox on thee for making me relive that one ;-)
If it is the game I am thinking of, it was even worse than that. The Giants only had to run out the clock, but for some reason chose a handoff which was bobbled. The Eagles (Herman Edwards maybe?) stripped the ball and took it in for the winning TD.