|Open the case and watch the CPU fan when you are cold starting the computer. See if that fan is turning like it does after the computer does start. If the CPU cooling fan does not turn fast enough, the computer will not boot.|
Try installing HWMonitor and use it to monitor the system temperatures and voltages. Maybe your power supply is failing and not providing the proper voltages. The 3.3V, 5V, 12V rails should all read within 5% of those values +or-. Get HWMonitor at the link below.
Blow out the crud. Use canned or compressed air. DO NOT use a vacuum cleaner. This can be messy so you may want to move the tower elsewhere. Be sure to also blow out the power supply from both ends.
After that, re-seat all the cables and cards. Snap the memory modules each in and out 4 or 5 times to burnish the contacts.