FLC OF AN MLG ACFT STOPPED THEIR DSCNT IN RESPONSE TO A TCASII RA JUST PRIOR TO ATC REQUESTING THEM TO CLB BACK TO PREVIOUS ASSIGNED ALT.

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CAPT FLYING; I'M FO; RUNNING RADIOS. ZJX GAVE CLRNC TO DSND PLT'S DISCRETION TO FL240 FROM OUR CRUISE ALT OF FL330. I ACKNOWLEDGED AND CAPT REPEATED. JAX CHANGED US OVER TO ZMA WHERE I CHKED ON 'FL330 WITH CLRNC TO FL240 PLT'S DISCRETION.' CTLR ROGERED AND SHE WAS VERY BUSY. SOON WE STARTED DSCNT AND MADE CALL 'LEAVING 330.' TCASII ALERTED 'TFC' AT 31500 FT; LOOKED AND SAW TFC 11 O'CLOCK. TCASII ALERTED 'REDUCE DSCNT.' CAPT FOLLOWED DIRECTION THEN TCASII ALERTED 'CLB; CLB.' CAPT FOLLOWED TCASII RA AND SMOOTHLY CLBED. THE OTHER AIRPLANE WAS IN SIGHT THROUGHOUT AND PASSED BELOW AND AHEAD 2 MI; 1500 FT VERT. CTLR TOLD US TO 'CLB AND MAINTAIN FL330' ABOUT THE TIME THE TCASII RA CAME UP. I RESPONDED AFTER CAPT HAD COMPLETED THE RA MANEUVER SO I COULD BACK HIM UP. I TOLD CTLR I HAD RPTED ON WITH PLT'S DISCRETION TO FL240. SHE REPEATED INSTRUCTION TO FL330. CAPT CLBED TO FL330. CAPT THEN CALLED CTR HIMSELF AND ASKED WHEN DSCNT COULD BE MADE. DIFFERENT CTLR CAME ON TO CLR US DOWN. SOUNDED LIKE ZJX AND ZMA HAD COM BREAKDOWN IN HDOF AND SOMEONE MISSED THE PLT'S DISCRETION CLRNC.