Can turboprop blades break the sound barrier?
Can it happen: Yes 

Does it happen: Generally no 

As a matter of the pure physics propeller tips can go supersonic and some times do. There is at least one case of this being done by design on the XF-84H which was built to be one of the fastest propeller planes. 



The main issue it faced was the noise generated by its supersonic prop. It is considered one of the loudest aircraft ever made as its prop was supersonic even at idle speeds. Due to the shock waves generated by breaking the sound barrier great care is often taken to ensure that propellers don't go supersonic.