Something just came to my mind:
It could always be a hardware failure. I once had a similar problem with a cheap computer from the internet, and it turned out the motherboard was responsible. Not every damage to a component results in total failure.

If at all possible try to replace the involved components (cpu, mb) and see if it changes anything. Maybe you have a nice friend that lends you his mb for testing or something...

Hmm a friend just bought a new i7 and mobo, so perhaps I'll ask if I can throw them in here to test it.

As I said earlier, I've been looking into getting a new mobo and CPU anyway, so if that is the case then that'll be a good upgrade.

And really that's the only thing I can think of that would be failing. The heatsink and PSU are brand new. My HDs are working fine. All the case fans work perfectly. The only things left are the CPU and motherboard.