ARTCC INTERFAC COORD ACR Y CLB THROUGH OCCUPIED ALT TCASII RA DSND HAD LTSS WITH ACR X TCASII TA RA CLB NON COMPLIANCE WITH ATC INSTRUCTIONS. SYS ERROR.

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ACR X WAS NBOUND AT FL370; ACR Y WAS WBOUND AT FL350. THE SECTOR WAS EXTREMELY BUSY AT THE TIME OF THE INCIDENT. ACR Y REQUESTED AND WAS GIVEN FL390. CONFLICT ALERT WENT OFF AND BOTH ACFT WERE GIVEN L TURNS AND ACR X WAS TOLD TO DSND TO FL350 IN AN ATTEMPT TO SWAP ALTS BUT IT WAS TOO LATE AND AEDP WENT OFF. SUPPLEMENTAL INFO FROM ACN 226561: I WAS WORKING MYS RADAR WHEN I ACCEPTED HDOF AND COMS ON ACR X. SME RADAR REQUESTED THAT I TURN ACR X TO A 270 DEG HDG AND DSND TO FL350 FROM FL370 AND TO EXPEDITE. I COMPLIED TO THE ABOVE REQUESTS. THE TFC FOR ACR X WAS AT FL350 CLBING TO FL390. ACR X RPTED AN RA AND TOOK ACTION. SUPPLEMENTAL INFO FROM ACN 226838: ACR X AT FL370; 190 NM S OF OKK; 350 DEG HDG; DIRECTED TO TURN TO 270 DEGS BY CTLR. JUST STARTING TURN; TCASII SHOWED ACR Y RIGHT IN FRONT OF US AT FL363 AND CLBING. TCASII TFC WARNING FOLLOWED IMMEDIATELY BY RA TO CLB. WHILE TURNING L TRIED TO CLB ACFT; OTHER ACFT NOW SHOWED FL365 ON HSI. DUE TO MAX ALT FOR ACFT; SUSTAINED A SHUDDER (EITHER MACH BUFFET OR STALL BUFFET). WHILE IN TURN CTLR THEN GAVE US A DSCNT CLRNC TO FL350. WE INFORMED HIM THAT WE HAD A TCASII RA CLB AND WERE TRYING TO HOLD WHAT WE HAD. AFTER APPROX 15 SECONDS; RA STOPPED AND WE DSNDED TO FL350. 2 PAX COMPLAINED OF INJURIES. ONE A SORE KNEE; OTHER SORE BACK. SUPPLEMENTAL INFO FROM ACN 226834: SEVERAL THOUGHTS AND SUGGESTIONS OCCURRED TO US AFTERWARD: 1) THE TCASII RESOLUTION DID NOT TAKE INTO ACCOUNT THE CAPABILITY OF THE ACFT. 2) MAYBE A TCASII CLB SHOULD BE TREATED LIKE A GPWS WARNING. 3) IT WAS DIFFICULT TO IDENT THE LOCATION OF ACR Y AS BOTH SCREENS WERE ON THE 320 NM SCALE. IT MIGHT BE A GOOD IDEA FOR 1 PLT TO BE ON A SMALLER SCALE. SUPPLEMENTAL INFO FROM ACN 226813: ACR Y CLRED TO CLB TO FL230. AT FL365 ATC CTLR ISSUED AN IMMEDIATE L TURN TO 180 DEGS. ACFT WAS HEADED 260 DEGS AT THIS TIME. TCASII GAVE AN RA. AUTOPLT WAS DISCONNECTED AND ACFT WAS DSNDED TO FL360. CONFLICT WAS CLRED ON TCASII. ACFT LEVELED OFF AT FL360. LATER CLRED TO FL390 AND BACK ON AIRWAY.