FLC OF 2 MAN WDB ALLEGEDLY TOOK DSCNT CLRNC FROM ZDC MEANT FOR ANOTHER SAME COMPANY FLT.

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FLT FROM POP - PUERTO PLATA DOMINICAN REPUBLIC TO NEW YORK JFK AT FL370. CHKED IN WITH WASHINGTON CENTER TWICE BEFORE ACKNOWLEDGEMENT. SECONDS LATER WASHINGTON CENTER CLRED OUR FLT TO CROSS A POINT 40 NE OF THE SIE VOR AT FL190. CLRNC READ BACK AND OUR AIRLINE AND FLT NUMBER CLRLY INCLUDED BUT NO ACKNOWLEDGEMENT BY WASHINGTON CENTER WHICH IS COMMON THESE DAYS AND ALMOST THE NORM. DSCNT STARTED AND AFTER DSNDING A FEW THOUSAND FT WASHINGTON CENTER CALLED AND SAID WE MUST HAVE TAKEN THE DSCNT CLRNC THAT WAS MEANT FOR ANOTHER ACFT OF SAME COMPANY. WASHINGTON CENTER THEN RECLRED OUR FLT AGAIN WITH THE SAME CLRNC; AND ALSO GAVE THE OTHER ACFT THE SAME CLRNC. THE FO AND I ARE QUITE SURE WASHINGTON CENTER USED OUR FLT NUMBER IN THE FIRST CLRNC; BUT NO MATTER; A HUMAN ERROR WAS MADE. THERE WAS A TELEPHONE FAILURE WHICH CAUSED THE LOSS OF NEW YORK CENTER FOR 4 OR 5 HRS. WE IN COCKPIT WERE QUITE BUSY WITH HOLDING CLRNC AND MESSAGES ABOUT THE PROBLEM AND POSSIBLE NEED TO DIVERT TO ANOTHER ARPT. WASHINGTON CENTER WAS PROBABLY BUSY DEVELOPING AND COORDINATING ALTERNATE PLANS AND RTES FOR ACFT INBOUND TO NEW YORK. ALTHOUGH FLT CONDITIONS WERE CLR AND THERE WERE VERY FEW ACFT (NO DEP FROM JFK) AND NO NEAR COLLISION THIS ERROR SHOULD NOT HAVE HAPPENED. BOTH ATC CTLRS AND PLTS MUST LISTEN MORE CAREFULLY TO CLRNCS AND READBACKS. CTLRS MUST ACKNOWLEDGE ALL CALLS AND IN PARTICULAR ACKNOWLEDGE READBACK OF CLRNCS.