Does the screen ever come up and turn blue, or does it simply stay with a black screen and no key light? If the screen stays black, but the keys don't light up, it might (for some reason) be going to the software upgrade standby screen. Normally, you get to that by pressing MENU and POWER at the same time. The screen turns on, stays black, and nothing else powers up. If you're simply pressing the power button, it may be reading input from the MENU button for some reason (other people have reported the keypad misbehaving with unintended input, etc).
Do you have an alternative power method? If you have the battery backing plate, try powering it from just a battery and see if it comes on?