Both Brady and Belichick are notorious control freaks and it would really only take them and a couple of equipment people.
It would help to compare the fumble rates of players before/after being on the Patriots against their fumble rates while on the Patriots.
It would be especially telling if they fumbled more after leaving since no NFL player wants to fumble, so if they knew of an edge that New England had developed in terms of coaching players to not fumble, they would bring it to the new team. Whereas if the ball was altered without their knowledge, they wouldn't know to take it to another team.
It's just at the least very suspicious timing that the fumble rate dropped so dramatically at that same point. It could be pure coincidence, but once you've already been busted for cheating (or if you want to take the most pro-Belichick position possible, misinterpreting a rule, one that was specifically noted by the NFL as being illegal), you kind of lose the ability to have others assume you aren't cheating (or misinterpreting the rules) again.