Maybe, I wouldn't know how to tell. My guess is that the motherboard is slightly faulty because it wasn't screwed into the case entirely, so it had a slight bend while the case was upright. Removing or adding RAM might just jiggle something somewhere in the right way to fix it.
It's one of these:
The metal fins can easily be detached, but the rest seems to be stuck together. When I tried to get to the motor by pulling on the blades I slightly cracked the top. It seems the circuit board is somehow attached to the casing and the motor, but I didn't want to mess with it more. I'm not sure it even works any more regardless of whether I can remove the hairs.
I will admit the inside of my PC was absolutely filthy. Since we have to add or remove RAM frequently I leave the side of the case off and it's lying on its side to prevent the motherboard from bending, so dust (and hairs) can easily get inside. I cleaned it out a bit recently but it's still not great.
So the interesting part is that when it fails to boot, the CPU fan refuses to spin but the CPU does get hot. I need to hold the power button to get it to turn off so I can fiddle with the RAM and hope it boots properly when I turn it on again. The case fan however does always spin, regardless of whether it boots or not.
The heatsink I have on it now is pretty similar to that one. It also has 8 pipes running through a bunch of metal fins, though it might be a bit smaller. It also allows a fan to be attached to the side of it, so I could buy a new fan for it if needed (or find the one I used to have on it).
I wanted to know if the heatsink was working well enough without a fan. Turns out that it works well enough during normal loads, but I should really avoid running it at 100% for more than 3 minutes or so.