LESS THAN STANDARD SEPARATION WHEN CORP JET CLIMBED TO ALT MENTIONED BY CTLR IN TRAFFIC ADVISORY.

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WE WERE BEING STEP CLBED FROM OUR DEP POINT BY CENTER DUE TO THE BUSY AIRSPACE. APCHING A LEVEL-OFF AT FL200 WE UNDERSTOOD ATC TO CLR US ON TO FL210. THE COPLT READ BACK THE CLRNC WITH NO REPLY FROM ATC. PASSING THROUGH 20700' ATC ASKED OUR ALT. THE COPLT ADVISED ATC OF OUR ALT; AFTER WHICH CENTER TOLD US THAT WE SHOULD BE AT FL200 DUE TO TFC AT FL210. I STARTED A DSCNT IMMEDIATELY. THE COPLT EXPLAINED TO ATC THAT HE HAD CLRED US TO FL210 AND CENTER'S REPLY WAS THAT HE WAS EXPLAINING TO US THAT HE WAS HOLDING US AT FL200 DUE TO TFC AT FL210. BEFORE OUR LEVEL-OFF BACK TO FL200 WE WERE RECLRED TO A HIGHER ALT. WE WERE LATER TOLD THAT WE SHOULD CONTACT THE QUALITY ASSURANCE OFFICE WHEN WE GOT ON THE GND. AS THE CAPT FOR THIS PARTICULAR FLT; I WAS FLYING THE AIRPLANE; AND WITH NUMEROUS SHORT CLBS AND LEVEL-OFFS; I WAS BUSIER THAN USUAL. ALSO; WHEN WE GOT WHAT WE UNDERSTOOD TO BE OUR CLRNC TO FL210 A PAX WAS ASKING US A QUESTION. THE CENTER WAS EXTREMELY BUSY WHICH ALSO CONTRIBUTED TO THE SITUATION. PREVENTING A RECURRENCE: ONE OF THE RESPONSIBILITIES OF ANY CREW IS TO MANAGE COCKPIT DISTRS; NO MATTER WHAT THE DISTR MAY BE. IT HAS BEEN OUR FLT DEPT'S POLICY TO CONFIRM ALL QUESTIONABLE CLRNCS WITH ATC AND BTWN CREW MEMBERS. EVEN THOUGH WE DID NOT CONSIDER THE CLRNC QUESTIONABLE AT THE TIME; IT IS EVIDENT WE MISSED PART OF ATC'S MESSAGE. I ALSO BELIEVE ATC SHOULD REFRAIN FROM USING NONPERTINENT ALTS WHEN RELAYING MESSAGES TO CREWS; PARTICULARLY WHEN THE FREQ IS BUSY. (EG; 'LEVEL-OFF AT FL200 DUE TO TFC ABOVE;' INSTEAD OF 'LEVEL-OFF AT FL200 DUE TO FL210 TFC.') WHY WE DIDN'T RECEIVE A CORRECTION FROM CENTER ON OUR CLRNC READBACK I STILL DON'T KNOW. TO BE SURE; I'LL BE LISTENING TO CENTER INTENTLY AND DOING MY BEST TO IGNORE COCKPIT DISTRS IN THE FUTURE. SUPPLEMENTAL INFORMATION FROM ACN 81387: THE FREQ WAS EXTREMELY CONGESTED WITH FREQUENT BREAKS IN TRANSMISSIONS AS SOMEONE WOULD CUT IN. WE HAD BEEN LEVELED AT FL190 AND AT FL200 AND WHEN I HEARD OUT CALL SIGN AND FL210 UNDERSTOOD IT TO BE A CLRNC AND ACKNOWLEDGED; ANTICIPATING ANOTHER 1000' CHANGE. THE CTLR WAS SO BUSY HE MISSED THE ACKNOWLEDGEMENT. I RECOMMEND THAT WHEN ATC GIVE TFC ADVISORIES THEY DO NOT MENTION ALTS. JUST SAY; 'TFC ABOVE OR BELOW YOU;' AND CLOCK POS.