Computer starts up on its own???

I helped my brother purchase a new XP400 and so far its great. But he hit me this morning with  something I never heard of. He claims he turned off his computer last night(start, turn off computer, turn off) and it shut off. When he got up in the morning it was up and running. He claims also to have watched it shut down and it did not reboot. He also said he "thinks" this happened other times too.

 

I tend to think he either forgot to shut it off or for some reason the  computer rebooted like they do sometimes instead of shutting off. But before I tell him he's full of it I was wondering if there may be some program restaring the computer from a complete shut off to run an update or some other task. You know... is there some esoteric progam I haven't heard of at work here or maybe some ghostly thing happening?

 

BTW he's 70 and I'm 60 so it may be a case of brain rot on both our parts. :smileysad:

 

He is running win xp pro.  He has a dsl router but the rest of the network is not set up. The other machine is a one year old XP with winxp pro.