Is not the angry snarly engine note of a Pratt & Whitney Wasp radial engine in a Harvard (Texan to our Colonial Cousins) the result of the tips of the prop blades going supersonic? Sure I read / heard that somewhere!
I wouldn't have thought so. The reason jet engines were developed in the first place was because propeller driven aircraft were reaching the end of their usefulness as fighters, due to the fact they had hit a ceiling as far as speed was concerned
This is because once the propeller begins to reach the speed of sound (M1) the shockwaves that are formed extend along the blade, disrupting the airflow over it, thus destroying much of the lift (thrust) that the blade was producing. Therefore if the blade tips were supersonic, the aircraft would be producing a much reduced amount of lift, thus making the aircraft terribly inefficient
This is why the majority of prop aircraft (with the obvious exception of early aircraft) have constant speed, variable pitch designs.