A320 FLT CREW HAS A TCAS RA AFTER BEING CLRED TO CLB BY ATC.

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FLT ACR X WAS LEVEL AT FL300. ZDC CALLED AND SAID SWITCH TO 133.82. WE READ THIS BACK. JUST BEFORE WE STARTED TO SWITCH FREQ; ZDC CALLED AND SAID 'ACR X CLB TO FL320.' WE READ BACK THE CLRNC WITH THE FREQ CHANGE. ZDC SAID 'NO; ACR X; DO NOT CHANGE FREQS AND CLB TO FL320.' WE READ BACK CLB TO FL320. WE CLBED TO FL320 AND LEVELED OFF. IN ABOUT 1 MIN WE GOT A TCAS ALERT -- TFC! WE CHKED OUR DISPLAY AND SAW A CONFLICT AHEAD LEVEL ABOUT 15 NM OUT. AN RA FOLLOWED WITH DSND! WE DSNDED. ZDC NEVER WARNED US OF THE TFC. THE OTHER PLANE WAS ACR Y. WE WERE FLYING TO PBI FROM PHL. WE TALKED TO CTR AND WERE GIVEN A PHONE NUMBER. I CALLED ZDC ON THE PHONE. THEY EXPLAINED THAT THE ATC CTLR CONFUSED US WITH AN ACR Z. HE GAVE US THE CLB BUT THOUGHT HE WAS TALKING TO ACR Z. SUPPLEMENTAL INFO FROM ACN 695877: WE WERE AT CRUISE ALT OF FL310 FOR ABOUT 30-45 MINS. ZDC WAS TRYING TO CONTACT ANOTHER ACFT (I BELIEVE ACR X); BUT GOT NO RESPONSE. WE GOT A TFC ALERT ON TCAS AS CTR CALLED TO TELL US WE HAD AN ACFT AT OUR ALT. WE ACKNOWLEDGED AND SOON GOT AN RA FROM TCAS TO CLB. WE COMPLIED WITH THE RA CLB; TOLD ATC WE HAD THE OTHER ACFT IN SIGHT; AND THAT WE WERE CLBING. MAX ALT WAS ABOUT FL323 BEFORE WE RETURNED TO FL310. THE OTHER ACFT CAME UP ON FREQ AND SAID THEY HAD DSNDED PER THEIR RA AND THAT THERE HAD BEEN A HDOF PROB WITH PRIOR CTLR. CTR RESPONDED THAT HE BELIEVED THERE WAS A CTLR PROB AND THAT TAPES WOULD BE PULLED AND INVESTIGATED.