When I was fault finding on m/cs with faults like yours I would remove everything that was not needed to get it into the basic operating mode. So remove all but one of the ram modules, all the sata discs except the one with the OS on it and any other cards you have installed except the video card obviously (unless there's an on board graphics option as well) externally just have the keyboard plugged in and see if you can get it to boot consistently, if you can add a component and repeat until you find which one starts the problem, if you still cant boot then check over the motherboard connectors including all the card slots, ram slots and power connectors using a magnifying glass (a 50mm camera lens is a good substitute) looking for bent or distorted pins, cracked plastic bits etc. also check the edge connectors on the video card and the pins in the psu connectors for discolouration as that might give you a clue.