My RT8272 buck dc converter fried in my test charging circuit for arduino
If you are using a buck converter, your input voltage must always exceed your output voltage. As your cap discharges, the voltage will drop and the current will rise to meet the demand.

The datasheet for your Buck shows no information for input voltages lower then 4.75v, therefore any behavior when supplied a lower voltage may not be consistent, nor what you expect. It's possible that you killed the chip when you browned it out by continuing to pull current until the cap has nothing left to give. 

For your next test, I might look to use the EN pin to disable the buck once the voltage has dropped to a critical level.