Can all surge protectors wear out, and how would you know if this is the case with the surge protector you have?
It truly depends whether your surge protector contains MOVs or not. 

The wear mechanism for a MOV is that the clamping voltage DECREASES slightly every time the MOV is required to clamp a transient. Eventually, the clamping voltage decreases to the point where the MOV starts to conduct on the peaks of every AC cycle. The MOV will then burn up. 

FWIW - the early cube plug-in surge protectors manufactured by RCA had a clear plastic window on the side that showed the red MOV inside. A label on one side of the cube said something along the lines of; "If it's black, take it back."

More modern surge protector power bars have thermal cutouts mounted between the MOVs. If a MOV flames, the thermal cutout interrupts the incoming power.