According to my sources:
One thing that has been reported over the last week, as we've been learning about how footballs are treated and measured before a game is that before every game, a referee measured with a pressure guage each football to see if the pressure in each one is between 12.5 and 13.5 PSI.
But that apparently isn't always the case. As a matter of fact, from what I've been told, may times the refs don't test the pressure of each ball with a guage at all. Sometimes the refs hold the ball, squeeze it, briefly inspect it, then sign off on it. Next ball. It's never been a problem before. This is apparently a well known fact in the NFL.
Now according to my sources, The Patriots turned in their footballs to the refs at a pressure that was just below the allowable PSI.
If it's a situation where the refs DID use a guage, the refs would see that the balls were underinflated, and inflate them to their proper size. But in this case, the balls were approved and given back to the Patriots under-inflated.
Thus, the underinflated balls. The Patriots according to my sources played with league approved deflated balls.