A CORPORATE JET FLC THOUGHT THAT THEY WERE CLRED TO DSND TO 11000 FT; AND READBACK SAME; TWICE. ARTCC CTLR'S TWICE; FAILED TO HEAR READBACK ERROR; AND ACFT DSNDED BELOW THE ALT THAT CTR HAD ASSIGNED -- CONFLICTING WITH TFC.

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WHILE DSNDING INTO BIRMINGHAM; ZME GAVE US A CLRNC TO DSND TO FLT LEVEL 210 BY 80 MI SW OF VULCAN VOR; AND THEN; 'PLT'S DISCRETION' TO WHAT I BELIEVED WAS 11000 FT. I READ BACK CLRNC; AND SET 21000 FT IN ALT PRESELECT. UPON REACHING FL210; I RESET ALT PRESELECT TO 11000 FT. MEM INQUIRED ABOUT OUR ALT WHILE WE DSNDED THROUGH FL220. WE RPTED OUR ALT; AND WERE HANDED OFF TO ATLANTA. I CHKED IN WITH ATLANTA AND ADVISED I WAS DSNDING TO 11000 FT. WHILE DSNDING THROUGH APPROX 12000 FT; I OBSERVED TFC N OF OUR POS; AND COMMENTED TO THE FO THAT ATC DID NOT CALL OUT THAT TFC. JUST ABOUT THEN; ATLANTA ASKED US TO SAY OUR ALT. I ADVISED 11500 FT DSNDING TO 11000 FT. ATC ADVISED OUR CLRNC WAS SUPPOSED TO BE 13000 FT. I SAID NEGATIVE; I HAD READ BACK 11000 FT. CTR SAID THEY WOULD CHK THEIR TAPES; AND I SAID I WOULD FILE A RPT. SUPPLEMENTAL INFO FROM ACN 298506: WE WERE ENRTE IFR; FROM HOUSTON (IAH) TO BIRMINGHAM (BHM). MEMPHIS GAVE US CLRNC TO CROSS 80 NM OUT OF VULCAN VOR (VUZ); AT OR BELOW FL210; THEN DSND AND MAINTAIN 11000 FT (FROM WHAT WE HEARD AND UNDERSTOOD). I WAS THE PF; SO THE CAPT READ BACK THE CLRNC; AS STATED ABOVE. WE WERE OUT OF 11500 FT FOR 11000 FT; WHEN THE CTLR ASKED US; 'WASN'T YOUR ASSIGNED ALT 13000 FT?' THEY HAD THOUGHT WE WERE DSNDING TO 13000 FT; AND IF THAT WAS SO; WE MUST HAVE MISUNDERSTOOD THE FINAL CLRNC; AND THE CTLRS HAD NOT CORRECTED THE CAPT'S READBACK TO CLR UP THE MISUNDERSTANDING.