x220 wont power if left shut down and plugged in(edit)

[deckoff]

The is the most weird behavior I've met - the x220 (i5, 4287-59G) sometimes will refuse to boot. Pressing the power button does nothing. I will try with battery in or out, re-set the BIOS, and so on. Then, after half an hour random attempts the PC finally booted and works OK. I even use it to write this post.
This happens for third time in the last 24h - when the PC works for a while it reboots without a problem. If left for some time, I have pray to the god of boot to make it boot. It is that random, really.
I have added 60GB SSD in the PCI express slot.
I dual -boot iwth Ubuntu (the machine is Ubuntu certified, so this should not be a problem)
I left the machine plugged in with battery full every time I had this problem.
Please, share any ideas or advises you might have - should I send it for repair how to prevent this thing from happening.

EDIT: It seems the behavior only happens if I leave the PC shut down and the battery is already charged. And I have to leave the x220 connected for a few hours at least. The laptop will power if left unplugged and battery out for a few minutes.

[EOMtp]

... the x220 (i5, 4287-59G) sometimes will refuse to boot. Pressing the power button does nothing. I will try with battery in or out, re-set the BIOS, and so on. ...

Clarification please:
If you get as far as being able to reset the BIOS, then the power button turned on the machine and the machine has booted already. Do you mean that the operating system will not start loading after the machine has booted ..or.. do you mean that the machine is dead because the power button does nothing?

[deckoff]

I mean - I unscrew some screws, pull out the BIOS back-up battery for 30 secs and try to power on again.
The power button will not power the machine at all ( if I see any text on a screen I am never scared)
I use one of these pads with fan, and the fan is powered by a USB. I am almost 100% that I left this pad plugged into the yellow USB (on the right of the x220). I stopped doing that and I have not encountered this problem anymore.

[deckoff]

It seems the behavior only happens if I leave the PC shut down and the battery is already charged. And I have to leave the x220 connected for a few hours at least. The laptop will power if left unplugged and battery out for a few minutes.