I had always thought that when the ball carrier was on the ground, they were down and the play was over. Recently I have seen a fair number of plays such as interceptions or fumble recoveries where the player intercepted (or recovered) the ball and fell on the ground, and then immediately got up and ran for as many yards as possible.
How does this work? Is there a time limit for getting up to resume play?
Since you specified American football with no indication to which league you are asking about (although I presume NFL due to your scenario), I will cover drastically different rules for the same scenario in two different leagues.
One important thing to note is that the ball carrier with possession of the ball can fall to the ground...and get back up to resume the play if said player has not been touched by a defender, given the player does not declare himself down (see Rule 7, Section 2, Article 1-d).
This is covered in Rule 7, Section 2, Article 1-a of the NFL rulebook:
An official shall declare the ball dead and the down ended:
when a runner is contacted by an opponent and touches the ground with any part of his body other than his hands or feet. The ball is dead the instant the runner touches the ground...
Note: If, after contact by an opponent, any part of a runner’s leg above the ankle or any part of his arm above the wrist touches the ground, the runner is down.
Aside: I recall a playoff game in which Marvin Harrison fell to the ground after catching the ball, and since no defender touched him to declare him down, he got back up and ran for a touchdown.
When can you get up again after falling on the ground?
Immediately...at least with respect to making an effort to get back up to resume the play.
Is there a time limit for getting up to resume play?
Yes. This falls under declaring himself down.
This is covered in Rule 7, Section 2, Article 1-d-1:
An official shall declare the ball dead and the down ended:
when a runner declares himself down by:
falling to the ground, or kneeling, and clearly making no immediate effort to advance
Unlike the NFL, the ball carrier with possession of the ball cannot fall to the ground and get back up to resume the play. The ball is dead upon touching the ground by anything other than your hands and feet.
This is covered in Rule 4, Article 3-b in the NCAA rulebook:
A live ball becomes dead and an official shall sound his whistle or declare it dead:
When any part of the ball carrier’s body, except his hand or foot, touches the ground or when the ball carrier is tackled or otherwise falls and loses possession of the ball as he contacts the ground with any part of his body, except his hand or foot
Note the difference between NFL and NCAA rules. In the NFL, the ball is dead when the runner is contacted by a defender and touches the ground. In the NCAA, the ball is dead when the runner touches the ground or is tackled to the ground.
