Stupid question, but is it completely powering off? What I mean is when this happens you see your shutdown splash screens and it's completely gone. Or can you press any button and it powers back on? That would just mean it went into stand by mode.
I just have to ask because I remember getting a call one day from this woman at my company saying her laptop crashed whenever it connected to the wireless network. She tells me she has to pull the battery and power it back on and she keeps losing work. Come to find out it was just hibernating because when she used the wireless it wasn't plugged in (She'd be at lunch or a meeting) and she'd leave it idle for long periods of time until it went into standby and hibernated.
So I'm not trying to insult intelligence, just have to ask!
It does sound like this problem does involve stand by mode, it locks up in stand by when on low battery. Try out the registry hackin the thread FourTwenSesh posted.