At first he did, but not as much recently. The Colts beat the Pats 4 or 5 times in a row before the Pats finally got one.
Two of the biggest chokes in recent history were on Brady teams -- SB 42 and the AFC Championship against the Colts the year before. The AFC Championship choke was particularly gratifying for me since the Pats lost an 18-point lead and then the excuses started flowing from Boston. "Bu...bu...bu...but Peyton has all the offensive talent!" Yeah? What happened the following year once the Patriots addressed their "lack" of offensive talent? That's right, SB 42!
Oh, and before the Pats fan brigade chime in, yes, you can blame the Pats offensive line or defense all you want, but you can say the exact same thing in the games Peyton lost and was branded a choker. As the Pats have recently learned, and us Colts fans have been saying FOR YEARS, you can't win without a complete team and a decent defense. I find it amusing listening to Pats fans whining about their defense and blaming that because when Colts fans did the same, we were told we were "making excuses for Manning."