The +12v pin might be connected to a small 3pin transistor near the header. If somebody overloaded it then that transistor could be blown. A good sign of this is if you measure the header and find some wimpy voltage like 1.5v, and probably not even stable at that.

It's also possible that both headers are provided by a common transistor somewhere else on the board.

If you do conclude that you have a bad transistor, you can either replace it, or bypass it with a direct line to +12v.
If you bypass it and the circuit isn't well understood, then it's probably safest to cut off the original connection to that pin, just in case the +12v would backfeed into something it shouldn't.

Thanks for the suggestions. I'll see if I can get to it this weekend and find out if there is a short somewhere. Hopefully not. It's a very nice early-era Intel ATX board.