I too am having problems getting mine to turn on...
If I power down completely, let it sit a day or two, pressing the power button does nothing - quick press/hold it in - nothing.
I plug in the charger - again nothing - no LED - press or hold the power button - nothing.
Let it sit a few minutes with the charger plugged in - press the power button - VOILA! - it powers on.
It's not a low battery - today it still had 83% remaining.
I'm still on the firmware that came on it - Android 3.2.79 - wonder if updating to 4.0.7 might help?
I recommend that you read through the thread LIVE! - Firmware Update (4.0.7) - Report Bugs
before you do that. I am still on 3.2.80 for reasons I have expressed elsewhere in this forum, and mine also spuriously turns itself on and drains the battery. I have to plug it into the mains charger and wait for it to come to its senses - a bit of a bind, but not a show-stopper for me.