It's not the system as much as how they run their organization. The Colts problem is that they fail to plan for life without Manning...or most of their starters for that matter. Their backup QB is usually atrocious. The Colts are normally thin at most positions as well.
The Patriots, on the other hand, constantly draft and develop QBs behind Brady. They've got a number of failures and a few successes. For example, they were satisfied with Matt Cassell as a backup, but still spent a 3rd round pick on Kevin O'Connell in 2008.
The Patriots also tend to be able to plug holes across the board better than most teams. Aside: the Packers were tremendously successful at this last year.