SIMILAR CALLSIGNS ACR X CLB THROUGH OCCUPIED ALT HAD LTSS FROM ACR Z. SYS ERROR.

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WE WERE ACR X. ACR Y HVY WAS ALSO ON FREQ. WE CHKED ON ZDV ABOUT THE SAME TIME. CTR POINTED OUT THE SIMILAR CALL SIGNS; AND BOTH ACFT ACKNOWLEDGED. SOME TIME LATER; WE REQUESTED A CLB FROM OUR CURRENT ALT OF FL290 TO FL330. CTR SAID WE HAD OPPOSITE DIRECTION TFC AND TO EXPECT CLB CLRNC WHEN CLR. MINS LATER; CTR CLRED US TO CLB TO FL330 AND I STARTED A CLB OF APPROX 1500 FPM. LESS THAN A MIN LATER; CTR TOLD ACR Y TO TURN R (20 TO 30 DEGS) FOR TFC; AND ACR Y ACKNOWLEDGED. I VISUALLY ACQUIRED ACR Y'S (SAME DIRECTION; AHEAD; AND HIGH) TURN. AN ESTIMATED 15-20 SECONDS LATER; CTR DIRECTED ACR Y AN ADDITIONAL 30 DEGS OF TURN WITH EXCITEMENT/STRESS AUDIBLE IN HIS VOICE. THE CAPT TRIED SEVERAL TIMES TO CONFIRM WITH CTR THAT THE REQUESTED TURN MIGHT BE FOR US; BUT HE WAS BLOCKED. SEVERAL CALLS WERE MADE BTWN CTR AND ACR Y. CTR TRYING TO CONFIRM ACR Y WAS IN A TURN (OR TO INCREASE RATE). ANOTHER CTR CALL DIRECTED AN ACR Z TO STOP A CLB AND/OR DSND TO FL280 OR FL290 (I CAN'T BE SURE). ALL THIS RADIO TFC WAS GOING BACK AND FORTH IN ABOUT 15-20 SECONDS. AT SOME TIME DURING THIS EXCHANGE; OUR TCASII CALLED 'TFC' FOLLOWED VERY SHORTLY BY RA INSTRUCTIONS. I DISENGAGED THE AUTOPLT AND SET THE VVI IN THE GREEN ARC; APPROX 2500-3000 FPM CLB. THE RA RESOLVED BEFORE WE REACHED FL330. I THINK CTR CLRED US INTO CONFLICTING; OPPOSITE DIRECTION TFC; ACR Z. THE SIMILAR CALL SIGNS WERE DEFINITELY A CONTRIBUTING FACTOR; BECAUSE THE WRONG ACFT WAS TURNED INITIALLY AND THERE WAS NO TIME TO SORT IT ALL OUT. I DON'T THINK THERE WOULD HAVE BEEN A MIDAIR COLLISION; BUT TCASII PREVENTED A NEAR MISS. IT WAS NICE TO HAVE TCASII IN THIS SIT.