Inductive load shorts power supply
The above circuit in of itself would not cause the 5V ATX supply to drop out, as the max amount of power that this circuit could draw would be ~12mA. The circuit may have been built something a little different than the one listed above and may be shorting out, test it with a benchtop supply before using it on an ATX supply.

The ATX power supply probably has overcurrent, and/or undervoltage protection. Undervoltage protection is the most likely, if the supply dips below a given voltage (say 4.7V) for an extended amount of time, the supply shuts down. 

If you're looking to dim the LED slowly and turn it on slowly, a mosfet with an RC for the input might be best. 

Like this: