A problem I've always had with reading too much into 3rd down completion percentage is that it rewards defenses that are feast-or-famine on 1st and 2nd downs and can punish defenses that are more consistent.
The 2003 Pats had the best defense of the Brady-Belichek era, and were a respectable-but-not-superlative 7th in 3rd down conversions. Meanwhile, in that year, the #1 team in 3rd conversion %, the Tennessee Titans, were ranked 20th in yards allowed per game and 21st in points allowed per game. A telling factor in this: the Titans forced 184 3rd down attempts, the second lowest that year, while the Patriots forced a league-high 235 conversion attempts.
If you give up fewer first downs on 1st and 2nd down than other teams, you'll end up facing more 3rd down attempts, but they'll often be of the short-yardage variety. Still, you're better off forcing 6 conversion attempts in a drive and give them up at a 40% rate than you are forcing 2 conversion attempts and give them up at a 30% rate.
On Monday night, on the Dolphins' opening TD drive, the Pats forced 2 third-downs on an 84 yard drive. On the Dolphins' 3rd quarter FG, they allowed the Dolphins to go 87 yards before forcing a 3rd down try. So on the drives in which the Pats' D was on its heels, it managed to keep its pretty 3rd down rate by giving up too many 10+ yard plays on 1st and 2nd down.
Some of this almost certainly has to do with the change from the 3-4 to the 4-3 -- a defense based on penetration will force more negative plays and get into more 3rd and long plays, at the expense of the prodigious number of 3rd down tries the Pats' 3-4 defense has historically forced. It's a give-and-take.
