Based on a theoretical design for a six place aircraft I did as a class project which ended up with level cruise numbers predicted in the high transonic, and some collateral experience watching the design of the blade rows for GE's UDF engine, I'd tend to think it is possible (though not practical) to design a supersonic capable, propeller driven, aircraft; though I would tend to think that it would only break or sustain Mach 1 in a dive...
The limiter on maintaining propeller efficiency is actually keeping the tip speeds subsonic - hence the swept tips on a lot of modern high-speed propellers and helo rotors.