Per two individuals that I greatly respect, the Patriots have turned down offers of a second round pick for back-up quarterback Ryan Mallett. Twice.
Take this as you will, but that's incredible. Mallett was selected by the Patriots in the third round of the 2011 draft as the team believed him to be the best quarterback in the entire draft. It was a best player available move with the expectation that they either would earn a higher draft pick in the future via a trade, or they'd receive four seasons with a quality back-up.
If what Justis and Ethan are saying is correct, and I have no reason to doubt them, then the Patriots have turned down offers of a second round pick for Mallett on multiple occasions. That shows how highly the team values his service and ability.
My opinion is if they've actually fielded multiple offers, I wouldn't be surprised to see him moved during the draft. I've expressed that Mallett is a prospect that is greater than plenty of those in this upcoming draft. If a team would be willing to use a second on a lesser prospect, then Bill Belichick probably feels confident that he can gain an early second or future first for Mallett.
Edit: Denver apparently offered Tim Tebow and a second round pick for Mallett, just to put this rumor into perspective of when the offers may have taken place.
Edit2: Nick Underhill of Mass Live actually reported this a year ago, so I somehow hibernated through all of this. So I guess this was more common of knowledge than I thought.