Bill is perfect and all, but the Patriots could have improved their roster in a few ways.
The Patriots were never going to trade for offensive tackle Joe Thomas or wide receiver Calvin Johnson. If Bill Belichick and Nick Caserio were going to pull some strings, they would have found a player on an expiring, or nearly expiring, contract that offered a marginal improvement over what the team has on the roster.
We have no way of knowing whether or not players were available. However, if the player isn't considered a franchise player, there is a very high chance that teams would have been willing to trade for the right price.