WDB ACFT TAKES EVASIVE ACTION OPPOSITE TO CTLR INSTRUCTIONS DUE TO PROX OF ANOTHER WDB.

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ACR B FLT TO SJU-L1011 DEP FROM RWY 22R JFK; CLRNC WAS HDG 205 DEGS 3000 FT. AFTER AIRBORNE CLRNC GIVEN TO 5000 FT. DEP CTL GAVE L TURN 120 DEGS AND 5000 FT. DEP WAS ALSO TALKING TO AN ACR X 747 WHICH TOOK OFF PRIOR TO US FOLLOWED BY A JETSTREAM. DEP WANTED ACR X TO PROCEED DIRECT BETTY; BUT ACR X WAS DEVIATING DUE TO WX. DEP THEN TOLD ACR X TO PROCEED DIRECT BETTY WHEN ABLE. WE SAW AN ACFT 6-8 MI AHEAD OF US WHEN WE WERE GIVEN THE TURN TO 120 DEGS; BUT HE DID NOT SHOW ON TCASII. AFTER LEVELING AT 5000 FT FOR APPROX 30 SECONDS; DEP TOLD US TO MAINTAIN 5000 FT; AS IF HE THOUGHT WE WERE NOT. WE REPLIED WE WERE LEVEL AT 5000 FT. AS WE WERE NEAR TO INTERCEPT JFK 139 DEGS RADIAL; DEP CLRED US DIRECT SHIPP. DEP THEN CLRED US CLB TO 13000 FT. AS WE WERE PASSING 5500 FT DEP TOLD US LEVEL OFF AND TURN L TO 060 DEGS. I MADE AN ABRUPT LEVEL OFF AND QUICKLY STARTED A L TURN. ALL THIS TIME WE HAD BEEN IN AND OUT OF CLOUDS WITH LIGHT RAIN. ABOUT THIS TIME I SAW A LARGE ACFT (I BELIEVE A 747; PROBABLY ACR X) SLIGHTLY ABOVE US AT ABOUT 1-2 O'CLOCK LESS THAN 1/2 MI MOVING R TO L. I ALSO SAW A SMALL ACFT BELOW AND BEHIND US (3-4 O'CLOCK). CTLR THEN ADVISED US TO TURN R DIRECT SHIPP AND CLB TO 13000 FT. WE ADVISED UNABLE DUE TO TFC. CTLR SAID TFC WAS NOT CONFLICT; THAT IT WAS AT 4000 FT AND DSNDING. TFC WAS CLRLY NOT AT 4000 FT OR DSNDING. I JUDGED THAT CONTINUED FLT IN DIRECTION CTLR HAD DIRECTED WOULD PROBABLY CAUSE US TO HIT THE LARGE ACFT; SO I ADDED MAX PWR; TURNED HARD R AND DID RAPID CLB TO GO ABOVE AND BEHIND OTHER ACFT. AFTER THIS MANEUVER WAS STARTED; TCASII ADVISED; 'MONITOR VERT SPD;' THE FIRST INDICATION OF ANY TYPE FOR THE CONFLICT. WE PASSED ABOVE AND BEHIND CONFLICT ACFT. I ESTIMATE LESS THAN 500 FT. CAPT ADVISED ME THERE WAS A BLIP ON THE RADAR ALTIMETER AND THE TCASII HAD SHOWN RED DIRECTLY UNDER OUR ACFT SYMBOL. HE SAID HE HAD LOST SIGHT OF THE CONFLICT WHILE I WAS EVADING IT. I THINK CTLR HAD CONFUSED EITHER US OR ACR X OR BOTH WITH THE SMALL ACFT. MAYBE HE CONFUSED US AND ACR X WITH EACH OTHER. I'M SURE WE WOULD HAVE HIT THE 747 IF WE HAD NOT HAD A BREAK IN THE WX SO I COULD SEE HIM. SUPPLEMENTAL INFO FROM ACN 280262: ARR: ACR A; B767; COORD SEBOUND DSNDING WHILE DEP TFC ACR B H/L1011 WAS EBOUND CLBING. ACR A WAS TURNED EBOUND AS ACR B WAS TURNED ON COURSE SEBOUND. CLB AND DSCNT RATES OF BOTH ACFT WERE SUCH THAT WHEN LATERAL SEPARATION WAS LOST THERE WOULD BE VERT SEPARATION. ACR B LEVELED AT 5600 FT AS ACR A WAS DSNDING THROUGH 5000 FT. ACR B THEN DSNDED SAYING THE TFC WAS ABOVE HIM. ACR A WAS TOLD TO EXPEDITE DSCNT TO 4000 FT. ACR B DSNDED THROUGH 5000 FT BEFORE HE FINALLY STOPPED HIS DSCNT AFTER CONTINUAL XMISSIONS BY CTLR ON DEP POS THAT TFC WAS BELOW HIM STILL DSNDING. I FEEL THAT THE 2 CTLRS ON POS STATED (AR; DR) AND QUICK COMPLIANCE BY THE ACR A. CREW WERE INSTRUMENTAL IN AVERTING A MID-AIR COLLISION. I FEEL THAT EITHER TCASII OR PLT ERROR ON THE PART OF THE CREW ARE WHAT CAUSED THIS OPERATIONAL ERROR.