WDB FLC IS GIVEN CLRNC TO DSND THEN TOLD CLRNC WAS FOR ANOTHER COMPANY ACFT AND TO RETURN TO ASSIGNED ALT. READBACK HEARBACK PROB.

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JUST PRIOR TO REACHING HPW VOR; WE WERE SLOWED BY CTR TO MACH .76 AND VECTORED TO R OF COURSE. ABOUT 15 OR 20 MI FROM THE VOR; ZDC CLRED US DIRECT TO HPW AND THEN FPR. OUR CALL SIGN WAS COMPANY EO. IN OUR TURN; WE RECEIVED CLRNC 'COMPANY EO DSND TO FL310.' THE COPLT IMMEDIATELY SET IN FL310 ON THE FMC AND STARTED A DSCNT AND I READ THE CLRNC BACK 'COMPANY EO ROGER; LEAVING FL370 FOR FL310.' A FEW SECONDS LATER; ATC ASKED IF WE COULD TIGHTEN UP OUR TURN. I SAID NOT AT THIS ALT AND REDUCED SPD. AS WE STARTED OUR TURN TO A NORTHERLY HDG; I NOTICED VISUALLY AND ON TCASII; SBOUND TFC BELOW US. AT FIRST I THOUGHT WE WOULD TURN WELL INSIDE OF THE LOWER TFC; BUT I TOLD THE COPLT TO EASE THE RATE OF DSCNT AS THIS PICTURE DID NOT LOOK RIGHT. AT THAT TIME ATC SAID 'COMPANY EO MAINTAIN FL370.' I; IN TURN; ASKED 'YOU WANT COMPANY EO TO MAINTAIN FL370?' THE CTLR RESPONDED 'I DID NOT CLR YOU TO DSND' AT WHICH I REPLIED 'YES YOU DID; YOU JUST CLRED US TO FL310.' HE RESPONDED 'THAT WAS COMPANY EOB I CLRED TO FL310.' I RESPONDED THAT I HAD UNDERSTOOD THE CLRNC WAS FOR COMPANY EO; WAS SORRY FOR THE CONFUSION AND WE RAPIDLY CLBED THE 600 FT OR 700 FT BACK TO FL370. THE COPLT AND I BOTH SWEAR THE CTLR INDICATED COMPANY EO IN THE CLRNC AND HE DID NOT QUESTION MY READBACK. WE DID NOT KNOW THAT THERE WAS A COMPANY EOB ON FREQ. THE ONLY RECOMMENDATION I HAVE IS THAT CTR ADVISE EVERYONE THAT HAVE SIMILAR CALL SIGNS SO THAT THIS KIND OF ERROR COULD BE AVOIDED.