A family members iPhone 4 decided one day to literally just turn off (or so it appeared anyway) forever and wouldn't respond to any button presses or charging attempts. Nothing physically happened to the phone to produce this problem, it just shut off.
I figured I would swap out the battery with one from another 4 and see if it would work... AND IT DID! So after looking at the "dead" battery I decided to put that back into the phone from which I took the working battery from and tried to boot up the good phone and that worked too!?!
The only advice I can give here is to try removing and reattaching the battery and try booting the phone before deciding the battery is dead. My assumption of what happened is some small hardware hiccup that caused the buttons to become unresponsive or the battery somehow had a loose connection or something. I'm not totally sure what happened but I am glad that all phones are working without the need for a new battery.