connecting a LED to a PC case fan header
With a 3 pin PC fan header, it should be possible.


Pin 1 is GND.
Pin 2 is 12V power to the fan. If the motherboard supports fan speed control via a three pin header, this 12V pin is PWM controlled by the motherboard. 
Pin 3 is tachometer feedback from the fan. The fan pulls this pin to GND, usually twice per revolution. This pin has a pull-up resistor on the motherboard.


So yes, if you connect a LED with a current limiting resistor between GND and the 12V pin, it should vary in brightness according to the fan speed set in the software for that header.

With three pin headers, there is no standard PWM frequency, but I expect the frequency to be much too high for you to be able to percieve any blinking. What you will see is the LED getting dimmer as you "slow down the fan" in software.

And for completeness: 


With four pin headers, the 12V is constant. Pin 4 has a 5V 25 kHz PWM signal that tells the fan how fast to spin.
4 pin headers can often be configured to operate in 3 pin mode.