A B737-300 FLC IS ACCUSED OF ACCEPTING ANOTHER COMPANY'S FLT NUMBER AND CLRNC TO FL330 WHEN AT FL350 NW OF MSY; LA.

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ZHU PUT US AT FL350 (WRONG ALT FOR DIRECTION OF FLT). ADDITIONALLY; ANOTHER COMPANY ACFT WITH A SIMILAR CALL SIGN WAS ON THE FREQ AND BEING HANDLED BY THE SAME CTLR. ATC THEN TOLD US THEY HAD TO CHANGE OUR ALT TO FL370 OR FL330 FOR TFC. I CONFIRMED WITH CAPT WE WOULD WANT FL330. THEN ATC TOLD US HE WOULD DSND US WHEN CLR OF TFC 2000 FT ABOVE AT 2 O'CLOCK AND 2000 FT BELOW US AT 8 O'CLOCK. I CALLED THE 2 O'CLOCK TFC AND NEGATIVE CONTACT ON THE 8 O'CLOCK TFC. MOMENTS LATER; ATC INSTRUCTED US TO DSND TO FL330. I VERIFIED THE CLRNC AND CALLED DSNDING TO FL330 BY CALL SIGN. SHORTLY THEREAFTER (300 FT); ATC TOLD US TO RETURN TO FL350 AND FOR THE OTHER COMPANY ACFT TO MAINTAIN FL330. HE THEN TOLD US TO AGAIN 'DSND TO FL330.' LATER; ATC INFORMED US THERE MIGHT BE A POSSIBLE ALTDEV. I BELIEVE ATC CONFUSED THE CALL SIGNS. THE OTHER COMPANY ACFT NEVER RESPONDED TO ANY RADIO XMISSIONS AND I USED OUR CALL SIGN TO AVOID POSSIBLE CONFUSION. THERE WAS NO TCASII WARNING NOR ANY ACFT IN CLOSE PROX. IT IS ALWAYS DIFFICULT TO COMMUNICATE EFFECTIVELY IN A BUSY ENVIRONMENT AND SIMILAR CALL SIGNS ONLY ADD TO THE PROB. AS LONG AS THIS ADDED CONFUSION EXISTS; THE POTENTIAL FOR ERRORS WILL NOT BE MITIGATED; IN SPITE OF INCREASED VIGILANCE BY BOTH PLTS AND CTLRS. SUPPLEMENTAL INFO FROM ACN 469491: EBOUND; WE WERE FLYING AT FL330. ZHU REQUESTED AND WE ACCEPTED (WRONG WAY ALT) FL350 DUE TO ANTICIPATED TFC SEPARATION. ZHU NEXT TOLD US THAT WE WOULD HAVE TO CLB TO FL370 OR DSND TO FL330. WE PUT IN OUR REQUEST FOR FL330. THE VERY NEXT ZHU CTLR DSNDED US TO FL330; AND INFORMED US THAT ZHU WOULD BE LOOKING INTO A POSSIBLE PLTDEV.