ALT ASSIGNMENT MISUNDERSTANDING.

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WAS HANDED OFF FROM RDU DEP TO WASHINGTON CTR ON CLIMBOUT; ENROUTE TO SBND VOR; WASHINGTON ISSUED 1 OR 2 INTERMEDIATE LEVEL OFFS ENROUTE TO FILED ALTITUDE OF FL240. AS EACH LEVEL OFF WAS APPROACHED; FURTHER CLIMB CLRNC WAS RECEIVED ENDING WITH WHAT I UNDERSTOOD TO BE FL240 WHICH I READ BACK AND ACKNOWLEDGED. PASSING OVERHEAD SOUTH BOSTON; LEVELING AT FL240; HEADING APPROX 330 DEG; A DIFFERENT CTLR'S VOICE FROM WASHINGTON CENTER 'YOUR CLRNC WAS TO FL230.' THE CAPTAIN DISCONNECTED THE AUTOPILOT AND BEGAN A FAIRLY RAPID DESCENT TO FL230. CENTER GAVE IMMEDIATE RIGHT TURN TO 030; LEVEL 240. I REPLIED WE WERE ENROUTE BACK TO 230. MOMENTARY CONFUSION AS I TOLD CAPT TO TURN RIGHT TO 030 (WE WERE 330 DEG). I ASKED CTLR WHAT ALTITUDE HE WANTED US AT; WE WERE LEVEL FL230 TURNING RIGHT TO 030. CENTER REPLIED WITH INSTRUCTIONS TO REMAIN AT FL230; HEADING 360. IN SORTING THIS OUT WITH CENTER THE CTLR STATED 'THE CTLR I RELIEVED SAID HE THOUGHT HE HAD CLRED YOU TO FL230.' I RESPONDED 'WELL; WE READ BACK AND ACKNOWLEDGED FL240.' WASH CTR REPLIED 'OUR FREQUENCIES ARE SO BAD; IT'S ALMOST A GUESS WHAT WE HEAR.' I WAS AT A COMPLETE LOSS FOR WORDS! COMMUNICATIONS ARE SO VITAL; IT'S OBVIOUS THAT ONE OF US IS WRONG BUT UNLESS THE COMMUNICATIONS LOOP IS COMPLETE THIS IS LIKELY TO HAPPEN AGAIN.