T61 Power-On Problems

[VersaWizKid]

I bought this ThinkPad T61 (Type 7659-12U, manufactured August 2008) from eBay, advertised as fully working and shown booted to the BIOS. The laptop actually does boot to the BIOS, but that's only when you get it to power on.
I seem to have this problem of the power button doing nothing whatsoever at times. Other times I can get it to power on and boot up just fine. Even in those instance where I can power on the laptop, it only does so when plugged in. The battery works just fine and can run for a few hours, but it apparently can't power on the laptop by itself even being 100% charged.
Makes me wonder if there's a bad solder joint on the board or if there's a short anywhere, whatnot. Anybody else ever had a problem like this where your ThinkPad doesn't even try to power on?
This model has Intel graphics, so there should have been a much lower chance of failure happening than one with Nvidia graphics. Even if this did have Nvidia graphics that failed it would at least try to power on, but there would only be a lightshow.

[RealBlackStuff]

Remove charger, battery, HDD, wifi-card, RAM, then press and hold down Power button for ca. 20 seconds.
Plug in only charger, try to boot.
It should beep 1-3-3-1.
Switch off, add wifi-card, try to boot.
It should still beep 1-3-3-1.
Switch off, add HDD and RAM.
It should now boot up in BIOS or OS.
If the above beeps do not apply, as a last resort try another T40/T41/T42/T43 keyboard.
If still no luck, your motherboard is defective.