B737 FLT CREW ACCEPTS TKOF CLEARANCE INTENDED FOR ANOTHER ACFT ON INTERSECTING RWY. FLT CREW REJECTS AT LOW SPEED WHEN CONFLICT BECOMES APPARENT.

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UPON TAXIING TO RWY XXR AT ZZZ; WE WERE GIVEN CLRNC FOR TKOF AND TOLD TO TURN TO A HDG OF 350 DEGS AND EXPEDITE THROUGH 3000 FT. THE FO ACKNOWLEDGED THE CLRNC IN FULL; 'COMPANY NUMBER CLRED FOR TKOF RWY XXR 350 DEG HDG; EXPEDITE THROUGH 3000 FT.' THE PF; MYSELF; BEGAN THE TKOF ROLL. BOTH OF US SOON SAW ANOTHER COMPANY ACFT NUMBER ON A TKOF ROLL AT ABOUT OUR 2 O'CLOCK POS (FROM RWY YY). THE PF (CAPT) REJECTED THE TKOF AT APPROX 65 KIAS. APPROX 3 SECONDS LATER; ZZZ TWR SAID; 'COMPANY NUMBER STOP.' WE THEN TOLD ZZZ TWR WE HAD ALREADY ABORTED THE TKOF ROLL. WE STOPPED WELL SHORT OF THE RWY XXR/RWY YY INTXN AND TAXIED OFF THE RWY. BOTH MYSELF AND THE FO WERE CONVINCED WE WERE GIVEN TKOF CLRNC ALONG WITH A HDG AND ALT INSTRUCTION. WE HAVE NOT REVIEWED THE TAPES YET; BUT FEEL ATC RADIO CONGESTION AND GARBLED XMISSIONS MAY HAVE BEEN A CONTRIBUTING FACTOR. THERE WAS ANOTHER CARRIER ALSO RECEIVING LNDG CLRNC AS WELL AS OURSELVES; 'COMPANY FLT NUMBER' AND A 'COMPANY FLT NUMBER' -- THE FIRST TWO SYLLABLES OF BOTH CALL SIGNS BEING THE SAME. AFTER CONSULTING WITH OUR QRH; ONBOARD PERFORMANCE COMPUTER; DISPATCH; CHIEF PLT ON DUTY; AND AFTER WAITING THE REQUIRED BRAKE COOLING TIME FRAME; WE CONTINUED OUR FLT. WITHOUT HAVING LISTENED TO THE TAPES YET; I CAN NOW ONLY SPECULATE. I THINK CONSTANT VIGILANCE IS ALWAYS KEY. IN THIS CASE WHERE 3 ACFT; 2 OF WHICH HAD SIMILAR CALL SIGNS; AND ATC ARE INVOLVED IN CLOSE TIME PROX. SUPPLEMENTAL INFO FROM ACN 782132: I WAS POSITIVE THAT TWR CLRED US FOR TKOF AND ASKED TWR IF HE CLRED US. HE STATED NO AND SAID HE DID NOT HEAR OUR READBACK. HE STATED THAT AFTER HE CLRED COMPANY NUMBER (THE OTHER COMPANY FLT) FOR TKOF; THE READBACK WAS GARBLED AND THE ONLY THING HE HEARD WAS 'COMPANY NUMBER.' LISTEN CAREFULLY FOR CORRECT CALL SIGNS; CALL TWR IMMEDIATELY WHEN A REJECT OCCURS AND WHEN I SEE CONVERGING TFC ON A XING RWY; HAVE BETTER SITUATIONAL AWARENESS WHEN XING RWYS ARE IN USE. OTHER TFC IN THE AREA COULD HAVE HAD BETTER RADIO COMS (NOT STEPPING ON TWR AND ACFT).