B737 CLRED INTO POS ON RWY 2L AT BNA OBSERVED ACFT ON FINAL AND ELECTED TO CROSS RWY AS TWR ISSUED GAR TO LNDG TFC.

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I WAS CAPT ON COMPANY FLT; TO NASHVILLE. IT WAS OUR SECOND LEG OF DAY 1 OF A 3-DAY. WE HAD COMPLETED 1 LEG. WE WERE TAXIING TO RWY 2L BEHIND ANOTHER CARRIER MD80. TWR GAVE US A POS AND HOLD CLRNC ON RWY 2L. MY FO RESPONDED; 'COMPANY; POS AND HOLD RWY 2L.' WE COMPLETED THE BEFORE CHKLIST AS ANOTHER CARRIER MD80 MOVED OUT OF OUR WAY S OF TXWY B1. AS WE CROSSED THE RWY HOLD LINE; WE BOTH CHKED FINAL AND IT WAS CLR. I TURNED MY LIGHTS ON. STILL PERPENDICULAR TO THE RWY ABOUT TO TURN; MY FO LOOKED OUT AGAIN AND POINTED TO ANOTHER COMPANY JET COMING OUT OF THE CLOUDS/FOG. (WX ON ATIS WINDS 0203 2 1/2 BCFG FEW 3 55 BROKEN; ALTIMETER 30.15; RWYS 2L/C AND RWY 31 IN USE.) I BELIEVE THE 300 FT WAS MORE SCATTERED; ESPECIALLY ON APCH. UPON SEEING THE ACFT; I ADDED PWR AND EXITED THE RWY (ABOUT 1 MI AWAY). WE TURNED AROUND ON TXWY A. THE FO LET TWR KNOW WE EXITED THE RWY BECAUSE OF THE ACFT. THEY OFFERED NO EXPLANATION. SIMULTANEOUSLY TO OUR SEEING THE ACFT AND CLRING THE RWY; TWR SENT THE OTHER COMPANY AROUND AND GAVE A VECTOR FOR CLBOUT AND A 'SORRY.' WE WERE THEN CLRED FOR TKOF. I GOT TWR'S PHONE NUMBER FOR LATER. WE FLEW TO OUR NEXT CITY WITHOUT INCIDENT. UPON SHUTDOWN AT OUR ARR CITY; I CALLED BNA. THEY TOLD ME IT WAS COMPLETELY THEIR FAULT. THE TWR THOUGHT COMPANY WAS ON RWY 2C PER NORMAL COMPANY FLOW. THEY HAD CHKED THE TAPES AND THAT FLT HAD BEEN CLRED TO RWY 2L. NOW; I CHK MY FISH FINDER BEFORE TAKING THE RWY. OTHERWISE; I FEEL WE FOLLOWED ALL PROCS CORRECTLY.