|I doubt that using system restore has anything to do with you being able to eventually boot.|
Have you looked in Device Manager to see if there is any hardware not working properly?
Unplug the computer and then open the case. Re-seat all the wiring, expansion cards and memory. For the memory modules snap each one in and out four or five times.
Inspect the capacitors for bulging or leaking. The link below shows examples of bad caps.
Check for dirt/dust inside the case. If the CPU fan is not rotating fast enough the computer sill shut down.
Blow out using canned or compressed air. DO NOT use a vacuum cleaner. This will be messy so you may want to move the case. Be sure to blow out the power supply from both ends too.