I tested the "time off = hats" theory this week. Four days of no TF2, and when I came back I didn't get any item drops for over an hour, then when I played on a privately hosted server I got another Bonk! Atomic Punch (the drink, not the hat). So no truth to it, it seems.
I still have no hats other than the Cheater's Lament. My most-wanted are the Vintage Tyrolean and the Otolaryngologist's Mirror. I hold firm to my belief that there is a single dice roll that takes place the first time anyone plays TF2 that determines the frequency with which they will acquire hats. 5 sides say Never (and the die is weighted to usually fall on one of these), 1 side says Occasionally, and 1 corner (it must balance on the opposite corner to get this) says Every Freaking Day.