Two pins are all you need to power the fan, 3rd pin is to monitor fan speed, 4th is for PWM to control the fan speed.
Manually spin the fan to see if it rotates, also inspect the wire and make sure it is not torn.
Check the BIOS for any fan settings.
 
Grab a random fan and plug it in to either header to see if it works or use a 2 pin to molex adapter to see if the fan works.

Did you connect the correct two wires, and did you orient it correctly? If the fan has three wires, one will be gnd, one +ve and one tacho. You need to be connecting gnd and positive, and orient them correctly (I forget which is gnd and which positive, but one of them will the centre pin). Obviously that means it can be connected 4 different ways, but only one will work. I would think positive will be red and gnd black.

Well it can only be connected one way, because both headers have a plastic bit sticking out on one side so the connector only fits one way. And on the fan connector the pin holes are offset to one side, and there's two plastic notches on both side that slide along the plastic bits on the mobo headers. I can try to turn it around and slide it in the other way, but I'm afraid the plastic bits on the headers would break off, as the fan connector pushes it away if I slide it in that way.

If it only fits one way, then without a picture I can't suggest anything else. Forcing it on any other way could potentially do worse than just damage the connector. I'm not the expert I thought I was when I was younger - I know enough to be dangerous with some things 😀

It's a little hard to explain, so I added some pics. As you can see, the CPU fan is spinning just fine, but the case fan won't go in both connectors. I think it's probably just busted. Even tried to spin it up manually, but it won't take off.

JustRob wrote on 2025-01-27, 22:22:

As you can see, the CPU fan is spinning just fine, but the case fan won't go in both connectors. I think it's probably just busted. Even tried to spin it up manually, but it won't take off.

And I think this pictures shows where the problem is:
download/file.php?id=210985&mode=view

^ Looks like the wires to the fan are damaged. If you look where the motherboard ATX connector is on that picture, then look a little to the right and down as you follow the fan wires, you can see where they all "thin out" in a section. Now I could be wrong, but this is what the issue looks like to me from the picture... and the one that makes the most sense if the fan is otherwise in good condition and spins freely without too much friction.

So perhaps the wires crushed by something in the case?
Either way, shouldn't be too hard to re-do those wires.

Oh boy! That's messed up! Even my tired and failing old eyes can see that problem 😀

I think that's the picture playing tricks. I see it too on the picture, but I swear on the actual thing I can't spot or feel any thinning in the cables.

Anyway, I ordered a new Noctua fan to replace this old thing, guess we'll see if it works.

Well, if you're going to trash the old fan, might as well strip the wires closer to the fan (the black and red wires, at least) to see if it works if it gets power from there (just shove those wires into a molex connector with power.) That way, at least you'll know if the wiring really was faulty or not. Never hurts to have a spare fan.
I personally would never go for Noctua fans - too expensive for my taste, and I can get the same quiet thermal performance with cheap fans if I play just a bit to make sure they run at the right voltage where they don't make too much noise but still push OK amount of air.