Its entirely possible.
My 300 Watt PSU running my kids computer just stopped giving the juice on the 12 volt rail - or its transient time (from switch on to full juice) was just SLOW - so the hard drives would not spin up in time, and the motherboard never could see them on startup - but otherwise it POSTed just fine. I thought it was the motherboard, so swapped it out, and still had startup problems, though not as severe, since it was a more robust motherboard.
I replaced the power supply, and all was hunky dory.
The reason I think its his PSU, is that before mine went completely tits up, it would sometimes start, other times not, but a restart or a "warmup period" followed by a restart would solve the boot issue, and then it ran perfectly fine.... until the next shutdown/startup.
BB