ALTDEV ON CLB. CORRECT ALT READBACK AFTER ATC CLRNC; BUT WRONG ALT SET IN ALT ALERT. WHEN CHANGING ATC FREQ; THE ALT SET IN ALERTER WAS READ BACK AND ARTCC CTLR MISSED THE INCORRECT ALT; BUT INTERVENED WHEN ACFT WAS CLBING THROUGH THE ASSIGNED ALT.

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WE DEPARTED GSP AT XX55Z FOR MCO. THERE WAS A LARGE WX AREA SSE OF GSP. FO AT THE CTLS; CLBING THROUGH +/-9000 FT; DEP GAVE US CLB INSTRUCTION TO FL200. CAPT READ BACK CLRNC TO FL200. AT THAT TIME WE WERE ENCOUNTERING MODERATE TURB CLBING OUT. FL220 WAS ENTERED INTO ALT ALERTER WINDOW INSTEAD OF FL200. PASSING FL180; WE WERE HANDED OVER TO ZTL. CAPT INFORMED THEM THAT WE WERE OUT FL180 FOR FL220. ZTL RESPONDED; 'ROGER.' PASSING FL210 OUR TCASII WARNING ALERTED US OF TFC CONFLICT. WE WERE AT THAT TIME IN VMC CONDITIONS AND COULD SEE OUR TFC AT 2 O'CLOCK POS ABOUT +/-1000 FT ABOVE; FLYING NBOUND ABOUT 3-5 MI AWAY. AT NO TIME WERE WE IN DANGER OF COLLIDING WITH TFC. AT FL216; ZTL ASKED IF WE WERE LEVEL AT FL200. CAPT REPLIED; 'NO SIR; WE ARE CLBING OUT OF FL210 TO FL220.' IF THERE WAS A DISCREPANCY IN ALT ASSIGNMENT; WHY DID ZTL NOT QUESTION THE FACT WHEN WE HAD CHK-IN FL180 FOR FL220? AND WHY WERE WE ASKED; 'WHAT ALT WE WERE AT;' WHEN WE WERE FL216 FOR FL220. (+/-1500 FT ABOVE SUPPOSEDLY ASSIGNED ALT OF FL200.) I FEEL BOTH CREW AND CTR SHOULD ALWAYS VERIFY AND CONFIRM HDG AND ALT ASSIGNMENT AT CHK-IN AND HANDOVERS.