Getting drafted in the seventh round isn't exactly the greatest thing for players. Sure, it's great to say that you were drafted, but it's very rare that seventh rounders stick around. At least as an undrafted free agent you can chose a team where you might have a chance to compete.
Just look at one of the Patriots seventh round picks last year, Jeremy Ebert, he's already gone. Released today.
Still, that doesn't mean it's impossible by any stretch. You can also look to last year's draft to find the exception: Alfonzo Dennard.
The seventh round is a place where you bring people in to maybe just be a body in training camp, or where you pick up a guy who fell drastically. That's what the Patriots did with Dennard, whose story is so well documented by now. That seventh rounder ended up being a key piece for the Patriots defense, and will enter training camp as a starter. He was originally graded to be an early round talent, but fell due to other issues.
While it's not the same exact story with one of the Patriots' newest seventh rounders, DE Michael Buchanan from Illinois, it has the same basic plot.
Buchanan saw a drop in production in his senior year after losing weight due to having his mouth wired shut after getting into a fight. According to ESPNBoston, Buchanan still ended up being named Illinois' Outstanding Defensive player. He had 4.5 sacks that year, but 7.5 the year before. He would've gone much higher had he come out after his junior year but he still had as early as a third to fourth round grade in this draft.
Could Buchanan follow in Dennard's footsteps of going from seventh rounder to starter? It's unlikely, but based on his background, it would definitely appear possible.