I'm having this exact issue.
I got this last march from nintendo themselves at PAX East (I'm an enforcer there) and I've babied this thing like you would a small child. Now, when I go to turn it on, the blue light comes on, and the speakers pop, and the system shuts off.
I tried the adapter thinking it was the battery, nope still does it.
Tried reseating the battery (which was snug so i dont think it got loose). Didn't work at first. tried again a minute later and it worked.
Put the device into sleep mode, and noticed it turned off when I checked it a few hours later.
I'm pretty flat broke, is it possible to send this in for repairs since it's still under warranty for what is obviously a defect?
From what I can google, it's likely just a ribbon cable coming loose from the wear of the hinged design of the DS, but I'd rather not open up the device when I have a perfectly good warantty.