B757 FLT CREW LNDG MDPP-POP EXPERIENCED CONFLICT WITH SAME COMPANY B767 DURING LNDG.

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APCHING POP; WE WERE CLRED FOR THE VOR RWY 26 APCH AND CIRCLE TO LAND RWY 8 PATTERN. UPON ENTERING DOWNWIND TO RWY 8 AFTER THE VOR RWY 26; TWR ADVISED US THAT OUR TFC WAS ON THE ROLL; RWY 8 AND I ACKNOWLEDGED. WE THEN TURNED L BASE AND RECEIVED CLRNC TO LAND ON RWY 8. AT THIS POINT; WE RECEIVED AN RA TO DSND AND MAINTAIN VERT SPD; AND SIMULTANEOUSLY SPOTTED A COMPANY B767 ABOVE AND R OF US IN A L TURN. CLOSEST PROX WAS +400 FT. THE RA RESOLVED AND WE LANDED WITHOUT INCIDENT. WE HAD MONITORED THEM IN HOLDING AT 7000 FT. BOTH OF US; XXX AND YXXX; HAD SIMILAR NUMBERS WHICH I THINK CONTRIBUTED TO THIS INCIDENT. IT APPEARS YXXX MAY HAVE MISTAKEN OUR LNDG CLRNC FOR THEIRS. ARRS WERE BEING SEQUENCED FROM THE VOR APCH TO THE PUBLISHED CIRCLE TO LAND PATTERN; AND I'M NOT SURE HOW ACFT Y ARRIVED IN CONFLICT WITH US. THIS INCIDENT DEVELOPED WITHOUT ANY INTERVENTION FROM TWR OR APCH CTL; WHICH IS DISTURBING. SUPPLEMENTAL INFO FROM ACN 624952: WHILE OUTBOUND IN THE HOLDING PATTERN AT 6000 FT OVER POP APCH CTL CLRED US FOR A VISUAL APCH BEHIND LNDG TFC. AS I STARTED THE TURN INBOUND I SPOTTED THE TFC ON SHORT FINAL AND ACCEPTED THE CLRNC. BECAUSE OF OUR ALT I STARTED IMMEDIATELY CONFIGURING. DURING THE TURN INBOUND WE WENT N OF THE RWY EXTENDED CTRLINE AND THE CTLR CLRED US FOR A L BASE. ANC CLRED US TO LAND. BECAUSE OF OUR ALT I ELECTED TO FLY THROUGH THE FINAL APCH COURSE SBOUND TO ASSIST IN THE DSCNT AND SPD REDUCTION PROCESS. UPON TURNING BACK TO THE N TWR CLRED ACFT X FOR LNDG. AT THIS TIME I WAS IN A L BANK TURN TO THE N PREPARED TO TURN R ON THE FINAL APCH COURSE AND LAND. WHILE STILL IN THE L TURN I SAW ACFT X DIRECTLY IN FRONT OF US TURNING FINAL FROM THE N AND I CONTINUED THE L TURN TO PASS ABOVE AND BEHIND ACFT X; SEPARATED BY APPROX 2000-3000 FT HORIZLY AND 300-500 FT HORIZLY. ABOUT THE TIME I STARTED THE EVASIVE MANEUVER WE RECEIVED A TCASII WARNING AND RA. I CONTINUED THE L HAND TURN A FULL 360 DEGS AND ROLLED OUT ON FINAL BEHIND ACFT X AT WHICH TIME TWR AGAIN CLRED US TO LAND.