Off the top of my head, Moss comes to mind in a very similar situation as a player coming off a one year prove it deal. He apparently wasn't entirely happy with the deal they had on the table and tried to leverage himself the first few days of FA before signing. Vinatieri was tagged for a week once and then signed a three year extension. Neal was a UFA and was rumored wanting to play on the west coast but didn't get any offers - and then re-signed here for 4 years. I think Izzo was re-signed as a UFA.
I would suspect that it would be similar for most teams. A player has reached UFA becuase a.) the team doesn't want him back or b.) the team wants him back, but not at the price the player is asking for. In both cases, once the player hits UFA it would be unlikely that they would return, particularly a good player. I could see how some back-ups and JAGs may go to UFA and then come back once they see how the market views them, but once a good player hits UFA there is usually a team willing to pay the price. I think Gaffney was a UFA after his first year here and then signed on for another year.
I remember when Shawn Alexander was the next great RB and Seattle couldn't get a deal done and they tagged him and everyone predicted they would never get him signed long term as a result, only he signed a long term deal the following January that Seattle immediately lived to regret...
Overpaying to acquire or retain talent isn't some magic wand...just ask Redskins fans, or Seahawk fans, or Panthers fans, or Eagles fans, or fans of lots of teams who have and haven't won a damn thing this decade...
We haven't been to 4 superbowls and won 3 in this decade because BB is a stubborn value driven knucklehead who doesn't have a clue what it takes to win...
Bruschi was allowed to go to UFA at least once and I think maybe twice. At one point I believe he came close to signing with Green Bay but didn't because he wanted to stay in NE.