RWY INCURSION DURING A CALL SIGN MIX-UP DURING A 3 RWY OP AT BNA; TN.

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RWY INCURSION. WHILE TAXIING TO RWY 20C; I CALLED 'IN SEQUENCE FOR RWY 20C' (THERE WAS AN ACFT Y JET HOLDING SHORT; OFF TO THE SIDE). MY XMISSION WAS BLOCKED SO I REPEATED IT AGAIN. TWR IMMEDIATELY RESPONDED AND ASKED IF WE COULD ACCEPT AN 'IMMEDIATE TKOF.' I RESPONDED 'AFFIRMATIVE' AND TWR READ BACK A CALL SIGN THAT WE THOUGHT WAS OUR CALL SIGN -- IT SOUNDED LIKE OUR CALL SIGN -- AND CONTINUED TO GIVE A HDG AND DME CLRNC AND CLRNC TO TAKE OFF. HE ALSO SAID THAT TFC WAS ON A 4 MI FINAL AT THE BEGINNING OF THAT XMISSION. BOTH THE CAPT AND I SAW THE TFC (IT LOOKED LIKE IT WAS FOR RWY 20C BUT ACTUALLY WAS RWY 20R). I READ BACK THE TKOF CLRNC INSTRUCTIONS FOLLOWED BY 'CLRED FOR TKOF RWY 20C COMPANY NUMBER.' WE PROCEEDED ON THE RWY AND ACFT Y CALLED OUT ON FREQ AND ASKED IF WE WERE REALLY CLRED. TWR CAME BACK AND TOLD US WE WERE IN ERROR BUT CLRED US ANYWAY AT THAT POINT. WE ELECTED TO CLR THE RWY; COLLECT OURSELVES; AND SEE WHAT WE DID WRONG. A FEW MOMENTS LATER; WE WERE CLRED FOR TKOF. TWR SAID NOT TO WORRY ABOUT IT -- THEY WERE TAKING NO ACTION. WE ALL AGREED THE RADIOS WERE VERY BUSY (ALL 3 RWYS HAD AIRPLANES READY FOR TKOF). EXPECTATION THEORY SET US UP FOR A FALL. EVERYTHING SEEMED NORMAL AND FIT INTO PLACE. ACFT Y SAVED US BECAUSE I AM NOT SURE TWR SAW US IN THOSE FEW SECONDS. I DO NOT KNOW HOW I WOULD HAVE DONE THIS DIFFERENTLY OTHER THAN READING EVERYTHING BACK AND WAITING FOR A CORRECTION FROM TWR BEFORE TAKING THE RWY. THIS IS VERY FRUSTRATING BECAUSE I CAN EASILY SEE THIS HAPPENING AGAIN. MAYBE I SHOULD HAVE ADDED OUR CALL SIGN TO OUR 'AFFIRMATIVE' CALL. I DON'T REMEMBER IF I DID OR NOT. MAYBE BE MORE CAREFUL WITH MULTIPLE RWYS. SUPPLEMENTAL INFO FROM ACN 561734: IMMEDIATELY AFTER THE FO'S CALL; THE TWR RESPONDED WITH AN 'EXPEDITE TKOF CLRNC; TFC ON A 4 MI FINAL' AND STARTED WITH WHAT I THOUGHT WAS OUR CALL SIGN. TURNS OUT THERE WAS AN ACFT #BDXC ON RWY 20L. WE WERE COMPANY #BDXH. IF YOU CLIP THE FIRST AND LAST WITH TOO MANY RADIOS AT THE SAME TIME; AND 3 RWYS WITH TFC ON EVERY ONE OF THEM; THEN TROUBLE BREWS. I ASKED THE CTLR HOW MUCH PAPERWORK WE GENERATED AND 'NONE' WAS HIS REPLY. IT'S THE SAME STUFF; DIFFERENT DAY AND CREW. WHAT'S THE CURE? MAYBE ANOTHER FREQ FOR RWY 20L. WE CAN KEEP AN EYE ON EACH OTHER ON THE CTR AND R; BUT I DON'T HAVE A CLUE THAT MAYBE SOMEONE WITH A SIMILAR CALL SIGN IS ON THE R. WHAT ABOUT POSITIVE TFC CTL LIGHTS. GREEN YOU CROSS THE HOLD LINE; RED YOU HOLD SHORT. I WILL TRY MY HARDEST TO NOT MAKE THE SAME MISTAKE TWICE. WHAT SCARES ME THE MOST IS KNOWING IT WAS SUCH A PERFECT TRAP; THAT ON ANY DAY I MIGHT FALL FOR IT AGAIN.