His competition for playing time when he was healthy has 10 sacks.
Where do you want him to play?
That's not exactly the full case there.
First, his competition is more directly Mark Anderson, who has been playing left DE and left 3-4 OLB. Anderson came here struggling with the 3-4 OLB role. Anderson came here struggling in setting the edge on run plays. Cunningham played last year more often in the edge-setting role. Cunningham is also competing with Guyton and Ninkovich in the 4-3 OLB role, or even Shaun Ellis at 4-3 DE.
Further, those guys don't play every down. In fact, Belichick very much likes to rotate DE's and OLB's in for different situations. His far preferred strategy is a DE who is versatile enough to line up at DE or drop as a 3-4 OLB. Cunningham had the clear advantage over Ellis and Anderson in this role, based on both his size and his training in the Patriots' system.
Yes, Carter has clearly outperformed Cunningham. No question there. But clearly a spot opened up when Ellis failed to command the LDE position. The Patriots started the season with a lot of Ellis, and Anderson late. Now Anderson has been starting, and Ellis is sharing a few goalline snaps with Brace. Cunningham did not command that spot that opened, Anderson did.
Nor did Cunningham excel enough to be a top reserve, instead coming on the field on defense when Nate Jones, Julian Edelman, and Niko Koutovides came off, late in games. He had the opportunity to generate pressure at the end of games, as Anderson did in the first couple games to earn a lot more time.
Cunningham is not at all in the situation Brian Hoyer is, stuck behind a clear starter who will command all of the snaps. You could compare his situation much more to Woodhead. Woodhead was behind BJGE, Faulk was coming back, and two new guys were brought in; however Woodhead has some familiarity with the system. Woodhead rotates in, took over a kick return role, and contributes.
Unfortunately, Cunningham failed to do that. After one sack last year, he failed to make any contribution this year. Like Crabel and Redd and so many others, he failed to thrive in this system.
There was opportunity for him in this defense. I don't lay the blame at Carter's performance.