No. The NFL only allows one overtime period during the regular season - or whomever scores first. Whichever comes first ends the game.
If no one has scored at the end of the first overtime period, it is recorded as a tie and the game is over.
This is not true during the playoffs, however. During the playoffs the game continues until someone scores.
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If a Super Bowl game were to be tied after four quarters of play, then yes, they would play continuous overtime periods until one time scores. However, to date, no Super Bowl has ever gone to overtime.