LTT CREW THOUGHT THEY WERE CLRED TO CLB TO FL220. ATC THOUGHT FL200.

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WE WERE LEVEL AT 14000 FT. I HAD MY FO REQUEST HIGHER ALT FROM THE ZJX CTLR. MY FO CALLED THE CTR CTLR AND REQUESTED HIGHER ALT; AFTER ABOUT 20 SECONDS PAUSE; THE CTLR SAID; ACR FLT NUMBER CLB AND MAINTAIN FL220. MY FO DIALED FL220 INTO THE ALT SELECTOR THEN REPEATED BACK THE ALT OF FL220 TO THE CTLR. WE BOTH VERIFIED THE ALT FL220 IN THE ALT SELECTOR AND BEGAN A CRUISE CLB TO FL220. WE CONTINUED THE CLB AND AFTER PASSING FL212 THE CTLR SAID 'ACR FLT NUMBER STOP ALT SQUAWK; ALT DIFFERS BY MORE THAN 1200 FT; SAY ALT.' I RESPONDED BY SAYING 'WE'RE CLBING OUT OF FL214 FOR FL220;' AND SAID 'FL220 IS WHAT WE READ BACK TO YOU.' THE CTLR THEN SAID; 'ACR FLT NUMBER CLB AND MAINTAIN FL220.' NO FURTHER CONVERSATION OR ACTION OCCURRED OTHER THAN ROUTINE. WE WERE THEN CLRED LATER TO CHARLESTON; SC; DIRECT; AND LATER HANDED OFF TO THE NEXT CTLR. MY CONCERN IS; MAYBE WE DID NOT HEAR THE CTLR CORRECTLY OR THE CTLR DID NOT HEAR MY FO'S RESPONSE AS FL220. MY FUTURE RESPONSIBILITY AS A CAPT WILL BE TO LISTEN MORE CAREFULLY TO ALT ASSIGNMENTS WHEN SOMEONE ELSE IS TALKING. THAT WAY WE CAN VERIFY IT MORE AS A CREW.