I believe you can if your signal is strong enough relative to the real one, which can be pretty weak in certain areas (hence the need for this device).

My high school's math and science club tried to do this once to the school's clocks as a prank. We never got the transmission circuitry working, but I did write a program for my graphing calculator that computed and outputed the correct signal sequence for a given timestamp through its 2.5mm jack.

Edit to add: As I recall, our particular clocks synced at midnight or if you hit the reset button. I think that was to increase the chances of a good signal.