MD88 DEPARTS CRUISE ALT BASED ON CLRNC HEARBACK CLRNC READBACK ERROR. LOSS OF SEPARATION WITH SECOND MD88 RESULTS.

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DURING CLBOUT FROM ATL ON THE PNUT 3 RNAV DEP; WE WERE ASSIGNED 13000 FT AT OUR ALT. WE WERE LEVEL AT 13000 FT WHEN THE CTLR CALLED OUT TFC AT 3 MI OUT AND 1000 FT ABOVE US. WE COULD NOT SEE THE TFC. SOON WE DID ACQUIRE THE TFC VISUALLY AND RPTED THIS TO THE CTLR. THE CTLR RESPONDED 'MAINTAIN VISUAL SEPARATION WITH THE TFC -- 14000 FT -- CONTACT ZTL ON 134.5.' THE XMISSION WAS GARBLED AND THE FO; THE PF; THOUGHT THE XMISSION WAS A CLRNC TO CLB TO 14000 FT. AS THE CAPT CHKED IN WITH THE NEXT CTLR; I (THE FO) BEGAN A CLB TO 14000 FT. THE CAPT ASKED 'WHY ARE WE CLBING.' AT THIS POINT; I REALIZED THAT I MAY HAVE MISUNDERSTOOD THE CLRNC BUT WAS ALREADY PASSING 13400 FT. THE NEW CTLR HESITATED TO ANSWER US AND ASKED US 'DID THE PREVIOUS CTLR GIVE YOU THAT?' IN REF TO THE ALT. THE CAPT AND I THEN DISCUSSED WHAT THE PREVIOUS CTLR SAID AND WE WERE BOTH UNSURE OF WHAT EXACTLY IT WAS. THE TFC AT 14000 FT WAS MOVING AWAY FROM US AND DID NOT CREATE A CONFLICT. THE NEW CTLR SEEMED CONFUSED AS TO WHAT OUR PREVIOUS CLRNC WAS BUT IMMEDIATELY CLRED US TO CONTINUE OUR CLB TO FL220. WE CONTINUED ON THE RTE.