First, don't do that. Connecting fans (And other hardware) in ways they weren't meant to be connected is a good way for unforseen catastrophic problems to occur.
Do you have ANY other four pin PWM fans you can connect to that fan header for testing? If not, then I'd move forward with the idea that there is likely a fault in the fan itself. It's always possible that the PWM controller on the board or some part of it for that header have failed, but it's a lot more likely to be the fan. Might simply be time to upgrade to a better cooler. You can get a decent cooler, much better than stock, for not much more than the cost of a good fan.
What are your full hardware specs including motherboard, CPU, case, PSU fans, etc?