If you've got a friction shifter, and you probably do, yes you can change it to a six speed but changing the number of gears is not the end of it. You'll need to change the freewheel. The six speed freewheel will be wider than your five speed. With a little luck the change will will allow the rear wheel to fit back in between the drops on your bike. However, it's possible if the clearance is tight, that it won't fit. You can try seven speed, but the concern I previously mentioned may be more pronounced. Because you probably have friction shifting, your shifter will work but you will have to completely reset the rear derailer to handle the new, wider freewheel.