A C550 TAXIING TO A RWY FOR TKOF EXPERIENCED DISPUTED CALL SIGN CONFUSION WITH THE GND CTLR RESULTING IN A RWY INCURSION.

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C550 SII TAXI AT BDL. WHILE WE TAXIED FROM THE FBO TO THE RWY; WE WERE CLRED TO TAXI ON TXWY J; TXWY S; AND HOLD SHORT RWY 24. READBACK WAS PROPER AND CORRECT. A COMMUTER DASH 8 LANDED RWY 24 AND WAS EXITING RWY 24 AT TXWY S HDG OPPOSITE OUR DIRECTION. UPON APCHING THE RWY 24 HOLD SHORT LINE; WE (CAPT AND COPLT) BELIEVED TO HEAR '(OUR CALL SIGN) CROSS BEHIND THE DASH 8; CLRED TO CROSS RWY 33; TAXI TO RWY 24.' WE WERE STILL ROLLING AND CONTINUED TO CROSS RWY 24; AFTER READING BACK CLRNC VERBATIM. WE WERE THEN TOLD '(CALL SIGN) HOLD SHORT RWY 24.' I IMMEDIATELY STOPPED PLANE; SHORT OF RWY BUT BEYOND HOLD SHORT LINE. COPLT RESPONDED TO CLRNC '(CALL SIGN); HOLDING SHORT RWY 24; I THOUGHT YOU CLRED US TO CROSS RWY 24.' NEXT XMISSION FROM ANYONE WAS GND SAYING '(CALL SIGN); EXPEDITE ACROSS RWY 24; HOLD SHORT OF TXWY C.' WE DID EXACTLY THAT; AND ACKNOWLEDGED CLRNC. WE THEN WERE CLRED NORMALLY AND UNEVENTFULLY AFTER THAT. NO MENTION OF INCURSION WAS MADE. NOTE: A PLANE ON FINAL WENT AROUND AS WE WERE BEING CLRED TO CROSS RWY 24. IT HAD BEEN A LONG DAY FOR US; AND IT IS POSSIBLE WE MISUNDERSTOOD CALL SIGN. ANOTHER PLANE WAS WAITING ACROSS THE RWY ON TXWY C TO TAXI TO RWY 24. IT IS POSSIBLE THE CTLR SAID THE WRONG CALL SIGN. OUR COMPANY SPOKE TO THE CTLR 30 MINS AFTER THE INCIDENT AND THE CTLR SAID THEY WERE NOT GOING TO PURSUE THE MATTER WITH A VIOLATION. CTLR ADMITTED THAT 'THERE WAS A LOT OF CONFUSION AT THE ARPT' AND SAID 'WE NEEDED WORK ON OUR CLRNC READBACKS.' WE FEEL OUR READBACKS WERE EXACTLY AS WE HEARD THEM ISSUED. APPARENTLY THE CTLR DID NOT REVIEW THE TAPES BEFORE OUR COMPANY SPOKE WITH THEM.