We've done this many times when traveling to smaller Caribbean islands that have no big airlines going into them...and unless they're code share partners, then YES you must claim your own luggage & re-check it yourself with the next carrier. Be sure to leave yourself a long enough layover - because depending on the airport, after you claim your lugagge you may have to re-enter the main terminal, and go through security again.
The biggest downside I see to this, is that if you're not flying codeshare partner airlines, and something happens to your 1st flight (delay or cancellation) they have no obligation to help you get to your final destination, or to re-book you for a missed 2nd flight. They would only be responsible for getting you to your connecting destination, and by that time you may miss your 2nd flight on the other airline. If that 2nd flight was on time, technically the 2nd airline wouldn't have any responsibility to get you on another flight since you missed the flight. I'm sure this rarely happens, but of course it could!
