FLC OF A CPR MLG RESPONDED TO ANOTHER ACFT'S CLRNC AND DID NOT KNOW THAT UNTIL ARTCC RADAR CTLR INTERVENED DURING THEIR DSCNT. THEY IMMEDIATELY RETURNED TO ASSIGNED AND THE CTLR ASSIGNED VECTORS TO AVOID CONFLICTING TFC.

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OUR ACFT WAS GIVEN A DSCNT FROM FL390 DOWN TO FL350. WE WERE APPROX 160 NM FROM OUR DEST OF TRI; SO WE BEGAN A SLOW DSCNT APPROX 1000 FPM. AT FL380 ZDC ASKED OUR ALT AND WE STATED FL380 DSNDING TO FL350. IMMEDIATELY; ATC SAID 'MDT X; YOU WERE SUPPOSED TO BE AT FL390; TURN NOW R TO HDG 330 DEGS.' SO IMMEDIATELY WE CLBED RAPIDLY BACK TO FL390 AND BEGAN TURN TO HDG 330 DEGS. THEN ATC GAVE ACFT Y A TURN TO CHANGE HDG. AS WE WERE CLBING THROUGH FL385; ANOTHER ACFT POPPED UP ON OUR TCASII SCREEN THAT WAS NOT THERE AT OUR LOWER ALT. HE APPEARED TO BE AT OUR 11 O'CLOCK POS AND APPROX 10 MI AWAY LEVEL AT FL370. WHEN OUR HDG WAS GOING THROUGH 300 DEGS AND ALT FL390; ATC GAVE US A TURN BACK DIRECT PSK VOR HDG 259 DEGS. I TOLD ATC WE HEARD HIM CALL MDT X AND I REPEATED MDT X DSND TO FL350; AND HE SAID THE DSCNT WAS FOR ACFT Y. WE THEN ASKED IF THERE WOULD BE A RPT FILED AND ATC SAID NO; THAT THERE WAS NOT A PROB. I FEEL VERY STRONGLY THAT ATC JUST GAVE US (MDT X) THE DSCNT AND DIDN'T REALIZE IT UNTIL HE SAW THE CONFLICT ON HIS RADAR SCREEN. I DO KNOW FOR A FACT THAT I DID REPEAT MDT X CLRED DOWN TO FL350 AND ATC DIDN'T RESPOND. SO PART OF THE PROB WAS HE DIDN'T LISTEN CAREFULLY TO THE READBACK; IF HE CLRED ACFT Y INSTEAD OF MDT X. ALSO; FROM TALKING TO HIM; I THINK HE REALIZED HE MADE THE MISTAKE. I ALSO THINK OUR TCASII HAD A BLIND SPOT AND THAT IS WHY WE DIDN'T SEE THE OTHER ACFT UNTIL WE HAD CLBED 500 FT. INCIDENTALLY; THERE WAS NO TCASII ADVISORIES AND I DON'T THINK WE WERE CLOSE ENOUGH TO HAVE GOTTEN ONE EITHER.