It could be the power supply. The behavior suggests the initial load from a cold start is too high for the power supply to handle.
A capacitor in the power supply could be going bad.
Tricky one to catch though. If the HD is spinning up, it would suggest the 12V rail is healthy as the motor spinup normally hits the 12V rail (although on some of the newer drivers, they have started using the 5V rail).
If the power supply does not have individual support for each rail, unplugging the HD and powering it up might do the trick. If the BIOS post happens, then you can be pretty sure the power supply is about to go out.
If the power supply has individual rail support, it gets a bit trickier as removing power from the HD would not neccessarily allow the BIOS to post. You would need to remove any PCI cards, for example, and hope it is enough.
The fact the HD is spinning up says the motherboard relay is working as the power supply is definately kicking off.
If it were mine, I would think power supply is going bad, first.