ACR X NON ADHERENCE TO ATC CLRNC UNDERSHOOT ASSIGNED ALT HAD LTSS FROM ACR Y.

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ACR X DEPARTED MIA GOING SWBOUND OVER CANOA. ACR Y WAS NEBOUND ABOUT 15 MI SE OF CANOA. ACR X WAS ABOUT 25 MI NE OF CANOA INTXN CLBING TO FL350 LEAVING FL333 IN A VERY SLOW CLB. FOR ATC SEPARATION AT THAT ALT I NEED 2000 FT AND 5 MI; SO I TURNED THE ACR X TO A HDG OF 270 DEGS AND TOLD THE ACFT TO EXPEDITE TO FL350. AT THIS TIME; ACR X CLBED ALMOST 1200 FT; THEN STOPPED HIS CLT AT FL345 AND TOLD ME HE HAD ACR Y ON TCASII AND FOR THE NEXT MIN LEVELED OFF AT ALT. 3 TIMES; THE ACFT WAS TOLD TO CLB TO FL350 AND EACH TIME THE PLT REPLIED HE HAD THE ACFT ON TCASII. AFTER THE THIRD TIME; WE HAD INSUFFICIENT SEPARATION. THE ONLY WAY TO SOLVE THIS PROBLEM IS TO GET THE PLTS TO UNDERSTAND THAT THEY CANNOT USE TCASII TO SEPARATE THEMSELVES. CALLBACK CONVERSATION WITH RPTR REVEALED THE FOLLOWING INFO: RPTR EXPERIENCE 7 YRS RADAR; 8 YRS MIL. SEPARATION WAS 3.3 MI; 1400 FT. CTLR TRAINING WAS IN PROGRESS AND THE RPTR INSTRUCTOR HAD TAKEN OVER CTL OF THE POS FROM THE TRAINEE. THE RPTR BELIEVES THE ACR X WAS USING TCASII TO SEPARATE THEMSELVES FROM ACR Y. ALSO; RPTR THINKS ACR X TURNED OFF MODE C BECAUSE AFTER THE 3 HITS AT FL345 HE DID NOT RECEIVE MODE C ALT UNTIL THE ACFT WAS LEVEL FL350. FAA INVESTIGATION DOWNGRADED THE INCIDENT TO A NON-EVENT. SUPPLEMENTAL INFO FROM ACN 228094: ACR X WAS ISSUED A HDG AND AN EXPEDITIOUS CLB TO FL350. THE ACR X WAS 30 MI FROM HIS TFC OUT OF FL330 TO FL350 AT THE TIME THE CLRNC WAS ISSUED. IN 1 MIN; ACR X WAS AT FL345 AND HE APPEARED TO HAVE LEVELED OFF. WHEN ASKED TO VERIFY REACHING FL350; HE RESPONDED; NEGATIVE WE'RE AT FL345 AND WE HAVE TFC ON OUR TCASII. I BELIEVE THAT THE PLT WAS USING TCASII AS A MEANS OF SEPARATION AND NOT CONFORMING TO THE CLRNC ISSUED.