Laptop boots to blank screen

Hi all,
I received a laptop to fix today .. It behaves like it is overheating but
I'm not sure.

When the power button is pressed the laptop fan whirs for about 2 seconds
and then nothing happens the screen is left blank, no further activity.

If I pull the power cable out, push it back in, then press the power button
the same thing happens,

However if I power down by holding the power button for 7 seconds (leaving
the power cable attached); the fan whirrs for 2 seconds as before, but then
the laptop appears to cut the power (as if it is overheating) ... then 2
seconds later the fan whirrs on as if the laptop is rebooting. Nothing
appears on the screen ..

I did manage to boot as far as the windows logo by leaving the laptop in a
cold room for an hour and then trying.

So you see why it points to overheating .. but on the other hand when I feel
the heatsink with my finger it does not feel too hot at all! (I had another
laptop overheat, and it felt hot alright)

any ideas?