Not if you let the chain wear long enough that it ruins the cassette.

OP: when you get a chance, replace your chain and cassette at the same time. You should also inspect your chainrings because if your Rockhopper has a 22T granny ring as it is liable to wear away just as fast as a 22T cog on your cassette. Chainrings last (almost) forever on road bikes with 42/52T rings to spread out the load. But a tiny granny ring will wear out much faster. Next time keep a closer eye on chain wear.