If you have a scheduled task, such as a virus scan, it is possible that the computer would wake itself up from hibernation to execute the task that was scheduled for that time. It could have woken from standby if the table was bumped and the mouse moved. Its nothing to fret about - your computer isn't 'possessed' or anything like that.
I have an HP a710n desktop PC (Athlon XP, Windows XP Home SP 2).
Yesterday I tried to put it on Standby twice before going to bed, but it came out of Standby for some reason a second later. So I put it on Hibernate. The computer was obviously powered off, but not unplugged from an outlet.
And an hour later, I was woken up by the computer suddenly powering up - it somehow came out of hibernation on its own. I put it back on Hibernate, and this time unplugged it, and, as you can imagine, it didn't turn on again until I plugged it back in and pressed the power button this morning.
I haven't had any problems with Standby today.
Anyone have any clue how the computer could have turned on its own?