Can a power supply still provide a voltage if it is broken?
Base on the infomation provided it is hard to say if the power supply is broken or the computer motherboard is defective. I am assuming that the computer powers up with the another power supply. If this is true we can only assume that the power supply is problem. 

With the newer ATX power supplies there is circuity to generate a power good or power OK signal. Check if signal meets the ATX specifications. Also power supply rise time is also important. Here are some infomation from a previous post. 




Below are few references that might help you. 



References:


Troubleshooting PCB of an ATX power supply?
Using PWR_OK to Activate an ATX Power Supply Automatically