Machine only powers up the first time

So as a service tech, I have had to troubleshoot many a bad power supply or motherboard but here is an error I have not encountered yet. The tower I am troubleshooting is 64-bit HP compaq dx2250 microtower. I dont think it has vista on it any longer but it doesn't get that far. When mouse, keyboard, VGA display and power are plugged in and I press the power button, the fans spin up and the machine begins to power up but nothing comes on the monitor. (Monitor is on correct input setting and verified to work). Also note that there are no addition display chips installed, nor have there ever been any. So I power it down to verify RAM is seated, HDD plugged in, all cables and buses are correct and they are, but when I press the power button, there is no response at all this time no matter what I try. I even bypassed the button with another switch in case the relay was bad and no luck. So I left work, traveled out of town, and a week later I come back and give it another try. Here's what is strange, like a de javu, the machine does the same mysterious thing where it powers the first only, and no luck the subsequent times. It seems the power supply is ok, at least enough to power it the first time, and the motherboard LED is on. Obviously something has gone bad but what could it be? has anyone encountered a similar issue?
 
Three times actually.
  • One with a bad set of RAM modules.
    • The RAM modules were RMA'd. When I tried them in two completely different systems, they somehow locked up the system and removing the motherboard battery was the only way to get the system functioning again, even with the previous modules.
  • Two with a replacement motherboard.
    • The motherboard replacement would not boot with an i7-2600K. After calling the vendor they convinced me to try a lower level CPU so that I could flash the BIOS. Since all previous versions of the BIOS that I am currently running support the i7-2600K, I suspect it was a newer BIOS meant for Ivy Bridge support.
  • Three with a power supply failure.
    • The power supply failed in the middle of using the PC. During the process of rebooting, the PC would not do anything.
All three times the PC would try to boot but fail. On reboot attempt there would not be any signs of life. After disconnecting the battery and/or removing the power cord, the system would then try and attempt to boot once again. Long story short, it could be a number of things that could be wrong, don't rule any of them out.
 
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