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Broken wire. Poor connection. Fan issue. Port issue.
So, I would try a different spot on the main board, one of the numerous fan connector locations. Just make sure you plug something into the cpu fan connector as well, like a random system fan, so the system doesn't think the fan isn't plugged in.
Usually - it's a speed control setting that's dropping too low for the motor to pick up - motors usually have a low speed % threshold that they can't run underneath, or they stall or do things like what you are seeing. Try adjusting the minimum fan speed up if possible, or disable the auto-speed control and run it manual speed.
It may also be your energy savings settings kicking in - jmilcher is correct, if temp is low enough some systems will shut off fans entirely.
Or it's a failing motor/bearing and the fan needs to be replaced ASAP. That would also cause this.
Depending on the cooler, it could have a 0rpm function at low temps. I had an Arctic cooler on an FM2+ APU some time ago that would stay at fan off until temps got to about 42c, then it would start spinning. A BIOS fan profile change could make it spin up, but it worked well how it was setup.
As was stated though, it could be settings, a bad wire, or just a bad fan if you're not used to it shutting off completely.
MB fan controllers usually use temperature monitoring to set fan speed.
If the PC has been powered off for a while, then the CPU is cold (room temp) and fan controller will want to run fan at minimum RPM setting.
Some fan controllers are not smart enough to send a brief higher-power/PWM signal to the fan to start it up and then slow it down to the desired speed after a short delay.
Or, the fan being used wants to see a higher voltage (or PWM signal) in order to start up than the fan controller is sending to it under cold conditions and the fan is refusing to start spinning.
Try setting BIOS fan minimum speed to a higher value and re-test. Repeat until fan starts spinning every time.
Or, find a different fan that can work with that fan controller.