Players who are cut become free agents and can negotiate with any team. While the vast majority of these players do not play in the NFL again there are some who do and become very good players, even Hall of Famers. Maybe the most famous player who was cut was John Unitas, Hall of Fame quarterback, who was drafted by the Pittsburgh Steelers and released. He signed with the Baltimore Colts and became one of the most feared and respected quarterbacks of all times. So it can happen.
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An NFL player who gets cut has several options. If he wants to keep playing then he might contact other teams about interviews and/or tryouts. He's also going to contact his agent to help him find out who might be interested in him. He might get a job as a broadcaster, especially if he has a recognizable name. Lastly, he might go back home and get a job outside of football in his field of education or training.