MD80 CREW OBTAINS CLRNC TO DSND. CTLR DSNDED ACFT INTO ONCOMING TFC.

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WE WERE FLYING FROM MCO TO JFK AT NIGHT. WE WERE PROCEEDING DIRECT SIE VOR; AND HAD BEEN CLRED TO DSND TO FL280. AS WE LEVELED AT FL280; WE WERE AWARE THAT WE WOULD NEED TO START DOWN SOON TO MAKE A XING ALT ON THE ARR TO JFK. WE ASKED FOR LOWER; AND WERE GIVEN FL270. WE WERE TOLD THAT WE COULDN'T GO LOWER DUE TO TFC. ABOUT 300 FT FROM LEVELOFF THE CTLR TOLD US TO MAINTAIN FL270; WHICH WAS WHAT WE WERE CLRED TO ANYWAY. WE HEARD THE CTLR TELL AN ACR FLT TO EXPEDITE DSCNT. ABOUT THAT TIME WE SAW TFC ON OUR TCASII DISPLAYS; AT OUR ALT; COMING AT US FROM ABOUT OUR 1 O'CLOCK POS AT A HIGH CLOSURE RATE. TCASII CALLED TFC ALMOST RIGHT AWAY. THE CTLR GAVE THE ACR AN IMMEDIATE 30 DEG L TURN. RIGHT AFTER THAT; WE GOT A DSCNT RA; AND WE NOTIFIED ATC THAT WE HAD AN RA. HE TOLD US TO TURN 30 DEGS L IMMEDIATELY. WE ACKNOWLEDGED AND TOLD HIM WE WERE DSNDING BY THE RA. BUT SOMEWHERE IN HERE THE ACR FLT STARTED DSNDING TOO! HE WAS DSNDING MUCH FASTER THAN WE WERE; AND AT OUR CLOSEST APCH (3 MI?) HE WAS 1000 FT BELOW US. AFTER WE DSNDED ABOUT 700 FT; TCASII GAVE US A 'CLR OF CONFLICT' AND WE STARTED TO CLB BACK TO FL270. ATC THEN GAVE US CLRNC TO CONTINUE DSCNT TO A LOWER ALT. THE PROB WE HAVE HERE IS THAT WE FOLLOWED OUR TCASII COMMANDS TO DSND; AND YET OUR TFC WAS DSNDING ALSO. DID HE HAVE A DSND RA ALSO; OR DID HE HAVE NO RA; OR DID HE HAVE A CLB RA AND IGNORED IT? HE APPARENTLY HAD US IN SIGHT; SO MAYBE HE DECIDED TO IGNORE THE RA. WE NEVER FOUND OUT; BECAUSE THE CTLR WAS NON COMMUNICATIVE (MAYBE A LITTLE RATTLED BY THE WHOLE THING); AND WE CHANGED FREQS SOON AFTER THAT.