Absurd thing I didn't know. Thought all safeties would be two points.
And recovery in the example below would be a failed conversion attempt and no points awarded.
Yes, it's possible to score only one point in an NFL game
Here's an example of how a game could get to 34-1. First, the Falcons would score four touchdowns and convert four extra points. Then, after the fifth touchdown, the Falcons would line up for a one-point attempt, the kick would be blocked, the other team would recover and run the ball toward the far end zone (in an effort to score two points), the team would fumble short of the goal line, the Falcons would recover in the field of play, and the player with the ball would backtrack into the end zone, where he would be tackled for a one-point safety.