I had a similar issue with a laptop some time in the past. The cause of the issue turned was never determined, but:
On a Hewlett Packard 8200 Series Laptop:
When fully powered off, by power button, and Wake-on LAN disabled;
The computer would start itself again after a certain period of time, exactly ten minutes to the second.
I had BIOS password configured so no harm could be done, BUT:
This issue started immediately after I had engaged in some dubious and shady ‘enterprises’ (above and beyond your standard warez downloading, bit torrent etc.) so to speak upon the internet, over a year after I had first purchased the laptop. After multiple formats, and multiple probes with windbg, rootkitunhooker, and gmer, and every anti-virus and anti-malware product under the sun, NO issue was revealed. But a definite issue existed, for the computer, coincidentally, to start attempting to ‘turn itself on’ (a computer does nothing that is neither intended programming or an artifact [glitch] thereof). I never got to try the machine hooked up to a different internet connection to see if it changed the behavior.
This put a strong scare in to me. I changed ISPs, threw the old PC out, and bought a new one to be completely safe. I disengaged from any grey-area behavior, and the problem never resumed.
I know this measure seems a bit drastic - and may have been completely uncalled for and pure coincidence - and likely was since nothing ever came of it beyond my draconian corrective measures. But I was, and am, extremely paranoid - to the point where I tend to stay away from security software made by companies incorporated in the same country in which I intend to use it.
My advice to you, if you have been doing anything… shady, is to quit, and get a new PC. I leave you to draw the same conclusions I did.