Trading up -- 17
Trading down/into future -- 31
Trading a future pick for a present pick -- 0
The main perception that seems to follow Belichick is that he prefers to trade back to accumulate picks, and the statistics mostly reflect that. But it's not as if he doesn't trade up if there is an opportunity that presents itself, and at this time last year, a strong case had been building for him to do so based on the makeup of the roster and high total of selections made in the previous three years (33).
This year, our hunch would be consistent with what Daniel Jeremiah wrote on NFL.com -- if there is a draft-day trade, it would more likely be down the board/into next year than up.
Part of the reason is the forward-thinking in the team-building process in trying to set up more flexibility in the 2014 draft with multiple high selections. Another factor is the overall lack of picks the Patriots currently have in 2013, with five, which is the lowest total the Patriots would have made in a Patriots-based Belichick draft (and limits the ability to move up).