A CPR FLC IN A BE40 CLBED ABOVE THEIR ASSIGNED ALT WHEN THEY MISTOOK AN ALT ASSIGNMENT INTENDED FOR A SIMILAR NUMBERED COMPANY ACFT FOR THEIR OWN. THIS LED TO A POTENTIAL CONFLICT WITH AN UNIDENTED TWIN ENG ACFT.

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ON APR/XA/99 AT ABOUT XA50Z; NEAR THE SEMINOLE VOR; WE WERE ASSIGNED A HDG OF 310 DEGS AND CLB INSTRUCTIONS TO FL280. WHILE CLBING THROUGH THE MID 20'S THE CTLR INSTRUCTED US TO CLB TO FL350. I REPLIED BACK TO THE CTLR THAT WE WERE TO CLB AND MAINTAIN FL350 AND ACKNOWLEDGED MY TAIL NUMBER. SINCE THE CTLR NEVER REPLIED BACK THAT OUR CLB INSTRUCTIONS WERE WRONG; WE SET FL350 IN THE ALT CAPTURE WINDOW. WHILE CLBING THROUGH FL270 OUR TCASII ADVISED US OF TFC AT OUR 12 - 2 O'CLOCK POS AND ABOVE US. AT THIS POINT WE SLOWED OUR CLB RATE AND LOOKED FOR A VISUAL OF THE TFC. WHILE PASSING THROUGH ABOUT FL285 OUR TCASII GAVE US AN RA WARNING TO DSND. AT THIS TIME; WE STOPPED OUR ASCENT AND STARTED A DSCNT. AT THE SAME TIME ALSO WE ACQUIRED VISUAL WITH THE TFC AND RECEIVED A PANIC REPLY FROM THE CTLR TO DSND TO FL280 AND TFC AT 12 O'CLOCK. I REPLIED BACK TO THE CTLR CONFIRMING WE HAD A TCASII ALERT AND TFC IN SIGHT AND THAT WE HAD ALREADY STARTED A DSCNT. WITHIN A FEW SECONDS; WE LEVELED OFF AT FL280 AND THE CTLR ADVISED US OF ANOTHER ACFT AT 12 O'CLOCK AND FL290. THE CTLR STATED THAT WE WERE TO HAVE CLBED AND MAINTAINED FL280 BECAUSE OF THAT TFC AT FL290. I REPLIED THAT WE HAD THAT ACFT IN SIGHT AND THAT HE HAD GIVEN US INSTRUCTION TO CLB AND MAINTAIN FL350 AND THAT I HAD CALLED IT BACK TO HIM. I THEN ASKED THE CTLR; AFTER THE TFC PASSED BY; IF HE WANTED US TO CLB TO FL350 OR STAY AT FL280. HE REPLIED; 'STAY AT FL280;' SO WE DID. SHORTLY; WE WERE HANDED OFF TO THE NEXT CTLR AND WE CONTINUED ON THE FLT WITH NO OTHER PROBS. MY FEELING OF WHAT CAUSED THE PROB WAS THAT THE CTLR NEVER ACKNOWLEDGED MY REPLY TO CLB TO FL350 WHEN IT WAS FIRST ISSUED. I HAVE ALSO LEARNED THAT THERE WAS ALSO A SECOND COMPANY JET ON FREQ WITH SIMILAR TAIL NUMBER/CALL SIGN. THE CTLR NEVER ADVISED US OF COMPANY ACFT WITH SIMILAR CALL SIGNS ON FREQ. HAD THE CTLR ADVISED US OF THE OTHER COMPANY ACFT AND HAD HE REPLIED BACK; THAT OUR INITIAL RESPONSE TO CLB TO FL350 WAS INCORRECT; THEN THERE WOULD HAVE NOT BEEN AN INCIDENT.