Even though your bags are checked all the way through you will still need to pick them up at customs at JFK and have them cleared at JFK. If you see CLE on your bag tags, you will be able to drop them off at a special bag drop-off area located right after leaving customs; after that, you can go to your flight to CLE provided you have your boarding pass for that flight. On a side note, Airline Alliances don't mean much, they're just a big marketing scheme. All major airlines in the world comunicate with each other to a certain extent regardless of Alliance afiliation. So if you are connecting on a different airline, most airlines can check bags through on most flights. If they don't, in most cases you can insist strongly on it and it should happen. Have a nice flight.
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