FLC COUNTERS TCASII RA WITH EVASIVE ACTION MANEUVERS AT RA COMMANDS AND VISUAL SIGHTINGS.

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I WAS IN CRUISE FLT AT FL330 PROCEEDING DIRECTLY TO CNU VOR. AT APPROX XA05 Z; 100 NM S OF CNU AND ABOUT 30 SW OF TUL; THE FO AND I OBSERVED TFC ABOVE US AT FL350 AND ALSO BELOW US AT FL310. AS THESE ACFT APCHED; WE WERE TOLD TO TURN L 10 DEG ANOTHER ACFT THEN APPEARED ON OUR TCASII AT OUR SAME ALT OF FL330. WE WERE THEN GIVEN ANOTHER TURN 15 DEG L. AS WE MADE OUR TURN; A FOURTH ACFT IMMEDIATELY APPEARED ON OUR TCASII. WE WERE THEN GIVEN A FURTHER TURN ABOUT 40 DEG L TO A 270 DEG HDG. WE WATCHED THE TCASII AS THE FOURTH ACFT CONVERGED ON US. I BEGAN A TURN BACK TO THE N TO AVOID AN ACR Y JET RAPIDLY APCHING US. THE TCASII COMMANDED A CLB AS THE CTLR TOLD US TO TURN TO A 360 DEG HDG. WE MAINTAINED VISUAL WITH THE VARIOUS ACFT AT OUR ALT AND ABOVE US; SO WE TURNED AND ONLY CLBED 300 FT. I COULD NOT CLB BECAUSE OF THE ACFT AT FL350 AND I WAS ABLE TO SAFELY TURN VISUALLY AWAY FROM THE CONFLICT. ACR Y JET ALSO APPEARED TO TAKE EVASIVE ACTION. I CALLED MCI CTR AND DISCUSSED THE SIT. THE SUPVR SAID A RPT WAS GOING TO WASHINGTON AND THAT IT WAS A CTLR ERROR. I ASKED HIM IF THE CTLR WAS NEW AND TRAINING. HE SAID NO. WITHOUT THE FULL REAL TIME PICTURE THE TCASII GAVE ME; I BELIEVE WE WOULD HAVE HAD A MIDAIR COLLISION. CLOSURE SPDS WERE IN EXCESS OF 600 KTS AND WE CAME WITHIN 2.5 MI. ALL OF THIS OCCURRED VERY RAPIDLY EVEN AS WE WATCHED IT DEVELOP.