AFTER AN ALT EXCURSION; A PROBABLE MISUSE OF TCASII RA PROCS RESULTS IN A POTENTIAL CONFLICT BTWN 2 CONVERGING B737'S.

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WHILE CRUISING AT FL310; WE RECEIVED A TCASII ADVISORY 'TFC;' QUICKLY FOLLOWED BY AN RA 'CLB.' I IMMEDIATELY DISCONNECTED THE AUTOPLT AND AUTOTHROTTLES AND RESPONDED TO THE TCASII RA. AT THE SAME TIME; ATC GAVE INSTRUCTIONS TO TURN R 30 DEGS. SO AFTER THE 'TCASII CLB REQUIREMENT' WAS BEING COMPLIED WITH AND MET; I BEGAN THE R TURN AS WELL. WE HEARD HIS INSTRUCTIONS AGAIN. WE KEPT CLBING; TRYING TO GET AWAY FROM THE CONFLICTING TFC ON TCASII WHEN THE TFC REVERSED ITS PATH AND BEGAN CLBING TOWARD OUR FLT PATH. WE KEPT THE ACFT CLBING TO SATISFY THE TCASII. THE TFC CONTINUED TO GET CLOSER TO OUR ACFT; AND THE TFC PASSED OFF OUR L WING AT LESS THAN 1 MI. WHEN THE CONFLICT WAS OVER; OUR ACFT WENT FROM FL310 TO FL334 AND WE ADVISED ATC WE WERE RETURNING TO OUR ASSIGNED ALT OF FL310. AFTER LEVEL AT FL310; ATC ADVISED US THAT THE CONFLICTING TFC WAS A B737 WHO HAD CAUSED THE CONFLICT. ATC INFORMED US THAT THE ACFT DSNDED INTO OUR FLT PATH AND THEN REVERSED PATH AND BEGAN A CLB. THE CONFLICT ACFT BEGAN AND CONTINUED TO CLB TOWARD OUR PATH WHILE WE WERE CLBING IN RESPONSE TO OUR TCASII. I BELIEVE THAT THE CONFLICTING TFC 'IGNORED' THEIR TCASII WHILE TRYING TO RETURN TO THEIR ASSIGNED ALT.