Sadly that is not that case. The contract is per game. If they are cut midseason they forfeit the contract. NFL contracts are really nasty things. Most of the big valuations you see will probably never pay out. They can backload them so that a player with a 5 year $10mm contract will probably never see that amount. They'll get $1mm the first year, and the same for each year, with the remaining $5mm paid out at the end. If they don't make it that far they don't get it. It's not unheard of for a player to make it 50% of their time on contract and only see 20% of their pay.

I only know this because I have a family member who plays professionally. The NFL likes to present an image that they take care of their players. It's a business to make money (and a not-for-profit) and exists only to enrich themselves. If they can short a player out of his money they will do it.