Computer won't boot up after normal shutdown

Hi, I've been having this problem for a while now but it has been getting worse.
When I don't use my computer for some hours, overnight for example, it normally starts up well. However if I shut it down or put it to sleep and want to turn it back on immediately after it does not boot.
All the lights are on, the fans are spinning, but I just get a black screen, no BIOS screen, no beeps from the computer, nothing.
I've been reading about this and testing some things.
Things I've done:
Checked all the connections on the mother board;
Lights are on on the mother board (so it's getting power);
Lights are on on the graphics card (same);
Removed components like the graphics card and RAM (each slot one by one / tested different RAM sticks in different slots);
Disconnected DVD drive;
Connected the HDD to a different SATA port.

However the only sollution I've found so far is waiting it out.
What do you think this could be? Is the PSU about to bite the dust?
Could it be the HDD with the OS failing? (This hard drive was in my previous computer, it's been working for some 8 years now.)

Any ideas on how to find the root of this problem would be greatly apreciated.

Thanks