The only thought I have is that you have made certain it is fully charged?
Yes. But the guy at the UbreakIFix (Google's recommended service center) says they would want to rule out power and corrosion issues by trying to replace the battery and looking for water damage (and repairing it if they find it). There were signs of water damage a week before the phone went into the boot loop (or rather "stuck boot screen"), after it got a little water on it: it would shut down or reboot whenever I plugged in a USB cord. This went away a few days before the problem above happened, but there could still be a relation.