Here is the thing I can't get my head around....
Tom Brady throws a football for a living. Over the years. he has probably thrown hundreds of thousands of passes in practice, games, OTA's, etc. Like anything else, you practice with what you play with. Meaning, assuming he doesn't cheat, he is going to practice with a fully inflated ball. How in the world he didn't notice the balls were not inflated correctly is beyond me. Sure, the average person may not notice the difference, but I can guarantee a QB does. Just doesn't make sense to me.
A couple of possibilities (I'm not a quarterback but just off the top of my head...)
1. Inflation isn't the only variation from football to football. They can be cold, hot, fat, thin, wet, dry, frozen, overinflated, underinflated, slippery, tacky, worn, new, etc. The whole reason that teams are allowed to pick their own balls is because most quarterbacks hate the "out of the box" new footballs because they're very slippery before they're broken in. If every football was exactly the same
except for its level of inflation, it would be much easier to discern that one variable. Along those lines, balls deflate naturally over the course of the game and with the conditions. It's not like he's always throwing exactly the same "feel" ball. A wet ball in the first quarter against Miami in September is going to feel much different than a dry ball in the fourth quarter against Green Bay in November, even if they were relatively similar before each game.
2. People are vastly overestimating the difference in the "feel" of an underinflated versus properly inflated ball. It's not like one is squishy and the other is hard as a rock. It's a subtle difference, at most.
3. You're only playing with one ball at a time. If you handed Brady a 13.5 ball and then handed him a 12.0 ball, and asked him to compare the two, I'm sure he could tell the difference. However, if you handed Brady a SINGLE football and said "guess the PSI," I highly doubt he'd be very accurate. This is why the sequence they did on NFL Live was so idiotic. If you handle three footballs in sequence and are told
in advance "one is over, one is under, and one is right on," obviously you'd be able to tell much easier than if you took a single ball from under center and were asked "was that ball over, under, or right on?"
Think about it this way: Somebody puts three pairs of sneakers in front of you. One is size 11, one is size 12, and one is size 13. If you're asked "which is the 11, which is the 12, and which is the 13?" it'll be pretty easy because you just stick them in order by size. That's much different than someone putting a SINGLE pair of sneakers in front of you and making you guess what size it is.