FLC OF AN MLG OVERSHOT DSCNT ALT.

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DSNDING FROM 11000 FT MSL TO 8000 FT MSL; CLRNC WAS GIVEN TO TURN TO A HDG OF 160 DEGS; DSND TO 6000 FT FOR RWY 1R AT DULLES. BOTH MY COPLT AND I BELIEVE WE HEARD THE CLRNC AS STATED. I TURNED TO A HDG OF 160 DEGS AS THE COPLT READ BACK THE ABOVE CLRNC. PASSING 9000 FT MSL; ATC POINTED OUT TFC AT 10 O'CLOCK AT 12 MI; 7000 FT. PASSING 7700 FT; ATC POINTED OUT TFC AT 2 O'CLOCK AT 6 MI AND SAID WE WERE TO LEVEL OFF AT 8000 FT BUT SHOULD NOW EXPEDITE DSCNT TO 5000 FT. WE DID EXPEDITE TO 5000 FT. AS WE DID; ATC SAID NO TFC CONFLICT TOOK PLACE AND WE ASKED FOR A TELEPHONE # TO CALL WHEN WE GOT ON THE GND. WHEN I CALLED THE TRACON AND SPOKE TO THE SUPVR; HE SAID THAT HIS PERSONNEL WERE CERTAIN OUR CLRNC WAS ONLY FOR 8000 FT; AND COULD NOT REMEMBER MY COPLT READING BACK 6000 FT INITIALLY OR STATING THE ALT (6000 FT) WE WERE DSNDING TO WHEN WE VERIFIED INITIAL RPT OF TFC AT 1 O'CLOCK AT 12 MI. DURING OUR VERY PROFESSIONAL EXCHANGE ON THE TELEPHONE; HE INDICATED HE WOULD PULL THE TAPES AND I WOULD CALL BACK THE NEXT MORNING. OUTCOME: ATC SUPVR VERIFIED WE WERE ONLY CLRED TO 8000 FT INITIALLY BUT YES WE DID READ BACK 6000 FT ON THE INITIAL READBACK AND AGAIN WHEN WE VERIFIED THE INITIAL CALL OF TFC AT 1 O'CLOCK AT 12 MI. THE CTLR MISSED BOTH OUR INITIAL READBACK TO DSND TO 6000 FT AND THE REF TO OUR DSCNT TO 6000 FT WHEN WE VERIFIED THE TFC RPT. LESSONS/BEHAVIOR CHANGE: 1) PLTS ARE RESPONSIBLE FOR CLRNCS WHETHER READBACK CORRECTLY OR NOT.