ZMP ARTCC RADAR CTLR SAW A POTENTIAL CONFLICT BTWN AN ACR B727 AND AN ACR B737 CLBING THROUGH THE B727 ALT AND TURNED FOR SEPARATION. B727 GOT A TCASII RA DURING THE TURN AND FOLLOWED IT; BUT SEPARATION WAS LOST ANYWAY. CTLR RPTR ACCEPTED BLAME FOR THE LOSS OF SEPARATION.

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DURING INFLT CRUISE OP AT FL290 A POTENTIAL NEAR MISS SCENARIO OCCURRED OVER THE PAWNEE CITY; NE; VORTAC. A BRIEF SUMMARY OF ALL RELATIVE KEY EVENTS FOLLOWS: OUR FLT ACR X WAS ADVISED BY ATC (ZMP) TO INITIATE AN IMMEDIATE 30 DEG R TURN. ACR X WAS ENRTE FROM SFO TO CVG AND WAS PROCEEDING DIRECT (RNAV OMEGA) TO CAPITOL VORTAC. SPD WAS .74 MACH AND FLT CONDITIONS WERE IMC WITH LESS THAN 1 SM VISIBILITY (AND OCCASIONAL LIGHT TURB) DURING THE INITIAL STAGE OF THE TURNING MANEUVER; A TCASII TA ALERT OCCURRED FOLLOWED A FEW SECONDS LATER BY AN RA ALERT. THE WX RADAR WAS IN USE SO THE PROX OF THE TA/RA TARGET WAS DIFFICULT TO ACCESS ON THE SHARED RADAR/TCASII CONSOLE DISPLAY. THE ATC CTLR ADVISED US TO 'TIGHTEN OUR TURN.' I ADVISED MY COPLT TO DISENGAGE THE AUTOPLT; TO TIGHTEN THE TURNING MANEUVER; AND TO FOLLOW THE IVSI DISPLAYED VERT GUIDANCE. I OBSERVED A RED RA TARGET SYMBOL ALMOST SUPERIMPOSED OVER OUR POS WITH A PLUS 100 FT SEPARATION INDICATION. THE RA CONFLICT SIT WAS RESOLVED AS WE COMPLETED OUR COMMANDED DSCNT MANEUVER (200 FPM GREEN BAR) AND ROLLED OUT ON OUR NEW HDG. DURING OUR ESCAPE AND SUBSEQUENT (REPOSITIONING) MANEUVERING OUR ALT VARIED APPROX PLUS OR MINUS 450 FT FROM FL290 (INITIAL DSCNT FOLLOWED BY A CLB THROUGH FL290 DURING THE RECOVERY). WE ADVISED ATC OF OUR COMPLIANCE WITH THE TCASII RA GUIDANCE AND OUR ALTDEVS AS SOON AS THE REAL TIME SEQUENCE-OF-EVENTS PERMITTED RADIO COM. WE WERE ADVISED BY ATC THAT A 'LOSS-OF-SEPARATION' EVENT HAD OCCURRED. A NEW CTLR CAME ON AND CLRED US TO A DIFFERENT FREQ (AND TO ZKC). CONTACT PHONE NUMBERS WERE REQUESTED AND PROVIDED BY THE REPLACEMENT ATC CTLR. THE PLT FROM THE OTHER ACFT INVOLVED (ACR Y; A B737 KANSAS CITY TO SLC); MADE SOME COMMENTS TO ZMP SUCH AS 'WHAT WAS THAT ALL ABOUT?' WE DID NOT ESTABLISH VISUAL CONTACT WITH THE OTHER ACFT. I SINCERELY BELIEVE THAT TIMELY RESPONSE TO THE RA ALERT COMBINED WITH THE AGGRESSIVE TURNING MANEUVER PREVENTED A SERIOUS NEAR-MISS ENCOUNTER. PRIOR TO DEP I BRIEFED MY CREW THAT THE 'PLT' FLYING SHOULD IMMEDIATELY FOLLOW ANY RA RECOVERY GUIDANCE WITH THE OTHER 2 PLTS ATTEMPTING TO ESTABLISH VISUAL CONTACT WITH THE INTRUDER. SUPPLEMENTAL INFO FROM ACN 354115: THE B727 WAS BEING RADAR VECTORED TO CAP (CAPITAL) AT FL290. THE B737 WAS CLBING OUT OF MCI TO FL310 OVER PWE AND POINTS BEYOND. WHEN ACFT WERE 11-13 MI APART IT WAS NOTICED THAT THE B737 WAS ONLY OUT OF FL286 CLBING. TURNS WERE INITIATED AT THIS TIME. VALUABLE SECONDS WERE LOST WHEN THE B737 DID NOT TAKE HIS ASSIGNED HDG WHEN FIRST ISSUED. THE B727 RESPONDED TO AN RA AFTER THE ACFT WERE CLR (NON FACTOR). I HAVE BEEN AN FPL CTLR FOR NEARLY 10 YRS. I TRANSFERRED TO THIS AREA SPECIALTY RECENTLY AND HAVE BEEN CERTIFIED ON THE INVOLVED SECTOR FOR 1 MONTH. THIS ERROR IS MY FIRST. THE ONLY CAUSE THAT EXPLAINS THIS ERROR IS HUMAN ERROR; MINE.