Although it was alluded to earlier, keep in mind that, at least in the past, most of the money in these huge contracts is back-loaded. That means when a player signed a gigantic contract, most of the money was put into salary in the later years of the contract. Since NFL contracts aren't guaranteed, the chances of the player actually seeing that money is slim. The team can either re-work the contract (pushing the money even further into the future), renegotiate the contract anew, or just cut the player and not worry about it.
Particularly because of the higher cap, players (or more specifically, their agents) are now trying to seek more guaranteed money (ie, bonus money). I'm not smart enough to really figure out the ramifications of that, though.