ACR X NON COMPLIANCE WITH ATC INSTRUCTIONS POSSIBLE LTSS WITH ACFT Y. OPERROR. POTENTIAL CONFLICT.

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ACR X REQUESTED DEV N OF COURSE (R) ON CLB THROUGH 27000 GOING TO 31000; DUE TO CONVECTIVE ACTIVITY. CTLR CLRED US TO DEV R AS NECESSARY; THEN DIRECT; WHEN ABLE; TO NEXT FIX (RNAV DIRECT). GAVE US A FREQ CHANGE AND WE CHKED IN. APPROX 30 SECONDS LATER; NEW CTLR REQUESTED L TURN DUE TO TFC. WE ADVISED UNABLE; AND HE REQUESTED IMMEDIATE HARD R 368 DEG TURN. AS WE STARTED AUTOPLT TURN; CTLR TOLD US TO INCREASE RATE OF TURN. AS AUTOPLT WAS ONLY GIVING 15 DEGS BANK AT THAT WT AND ALT. THE CAPT DISCONNECTED THE AUTOPLT AND INCREASED BANK TO 30 DEGS. AT THE SAME TIME; CTLR REQUESTED IMMEDIATE RAPID DSCNT TO 29000. THE CAPT GAVE HIM 3500-4000 FPM. AT NO TIME DID THE CTLR RPT POS OR ALT OF CONFLICTING ACFT Y. WHEN I ASKED HOW CLOSE WE WERE; THE CTLR; VERY RELIEVED; SAID; 'WELL IT SCARED THE BLANK OUT OF ME.' I CAN ONLY ASSUME THAT THE HDOF FROM OUR PREVIOUS CTLR WAS NOT ENTIRELY COMPLETE; AS WE WERE DOING EXACTLY WHAT WE HAD BEEN CLRED TO DO. I ALSO THINK WE POSSIBLY COULD HAVE MADE A BETTER DECISION OF DEVIATING IF WE HAD REQUESTED THE CONFLICTING TFC POS OR THE CTLR HAD TOLD US IT'S POS.