if you have original PICKIT2 (not a clone) then chances of PICKIT2 malfunctioning are small. however, there is nothing preventing you from verifying that:
download software "PICkit 2 v2.61" from Microchip website. this is one handy program to have and should be used first to check if the programmer responds and does what it should. for example you can force signals on header so you can measure them with a multimeter (VPP, MCLR,...).
next thing is to make sure when connecting to target chip (PIC16F628A) you need to make sure that it is supported, and wired correctly.
if the wiring is correct, and target is ok, you can now read and write target, erase it, verify etc.
you should be able to read back same content that you write to target.