Yes, this is normal. If you replace your chain on a regular basis, you can ususally go through 2-3 chains per every cassette. If you wait too long, the worn chain will cause the cassette to wear as well (prematurely!)
Word to the wise, buy a chain checker (about $15 for a Park Tool) and replace the chain as soon as it hits the "RED" zone!
BTW, I learned the hard way as well!
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"Life is not a journey to the grave with the intention of arriving safely in a pretty and well preserved body, but rather to skid in broadside, thoroughly used up, totally worn out, and loudly proclaiming, "WOW, What a Ride!" - unknown
"Your Bike Sucks" - Sky Yaeger