Nah, the NFL isn't like MLB where small market teams can't compete. Every team ends up near the cap limit. In fact the teams that open up their pocketbook for free agents usually end up sucking. Washington spent the bank on Haynesworth and Al Davis of the Raiders overpays for lots of players. Both underachieve or in the Raiders case they just suck usually.
Good teams in the NFL aren't formed through free agency. They are formed through the draft and players want to be drafted as high as possible(to a crappy team) so they can bank the most.
I would be surprised if any team is losing money. If they are its their fault for failing at scouting/drafting.