MLG CREW CAUSES A TCASII RA AND THEN DOES NOT COMPLY WITH TCASII INSTRUCTIONS.

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INBOUND TO BNA DSNDING THROUGH FL280 TO FL270; FMA ANNUNCIATED 'ALT CAPTURE' AND FO ANNOUNCED 'CAPTURE.' AT FL272 AUTOPLT CONTINUED TO DSND; FMA DROPPED CAPTURE AND AUTOPLT INCREASED RATE FROM MINUS 1000 FPM TO MINUS 3500 ON FLT GUIDANCE SYS VSI INDICATOR. FO DISCONNECTED AUTOPLT AND PULLED BACK ON YOKE TO STOP DSCNT AND RETURN TO FL270. (DSCNT BOTTOMED OUT AT APPROX 26750 FT.) SIMULTANEOUSLY TCASII COMMANDED 'DSND; DSND' WE DID NOT FOLLOW COMMAND AS TFC WAS IN SIGHT AHEAD AND BELOW AND NOT POSING A PROB. TCASII FOLLOWED WITH 'MONITOR VERT SPD' AND THEN 'CLB; CLB' STILL REMAINED AT FL270 AS OTHER ACFT STILL IN SIGHT XING FROM L TO R. LATER INFORMED BY CTLR THAT OTHER ACFT HAD RESPONDED TO A TCASII ALERT BY CLBING. IT WOULD SEEM THAT IN CERTAIN CIRCUMSTANCES WHEN A TCASII COMMANDS A RESOLUTION; WHEREBY 1 ACFT HOLDS STATUS QUO DUE TO OTHER ACFT IN SIGHT; BUT OTHER ACFT RESPONDS TO COMMAND; THE SIT CAN AGGRAVATE RAPIDLY (AND EVEN REVERSE). SUPPLEMENTAL INFO FROM ACN 242769: ATC LATER ADVISED US OTHER ACFT AT FL260 RECEIVED A TCASII RA TO CLB. OTHER ACFT CLBED TO FL270 (FOLLOWING TCASII COMMANDS). TCASII CAUSED OTHER ACFT TO CLB INTO OUR ASSIGNED ALT WHILE OUR TCASII INSTRUCTED US TO DSND INTO OTHER ACFT'S ALT. COULD HAVE CAUSED SERIOUS SEPARATION PROBS IF WE HADN'T HAD VISUAL CONTACT WITH OTHER ACFT.