B767 FLC HAS TCASII DURING DSCNT FROM CRUISE ALT.

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PROCEEDING DIRECT TO NOR VERSUS FILED RTE UR10 CTL UZ706 NOR; FL360. PARIS CTL CLRED OUR FLT TO FL340 AND ASKED US TO MAINTAIN A MIN OF 1000 FPM IN THE DSCNT; IN THE SAME XMISSION THEY GAVE US A FREQ CHANGE TO MAASTRICH CTL. PARIS DID NOT RESPOND TO OUR READBACK OF THE CLRNC; SO WE SWITCHED TO MASS. THE FREQ WAS VERY CONGESTED. THE CAPT MADE 4-5 XMISSIONS; BUT GOT NO RESPONSE. WE COULD HEAR OTHER ACFT AND MASS XMITTING IN A MALE VOICE. WE HEARD A BLOCKED XMISSION TELLING SOMEONE TO STOP AT FL345. FO LEVELED AT FL345. STILL; NO LUCK WITH MASS WITH ALL 3 CREW MEMBERS XMITTING AND CHKING CONNECTIONS. R RADIO WAS ALSO TRIED. TCASII SHOWED CONVERGING TFC AT 2 O'CLOCK AND FL344. CAPT TOLD THE FO TO CLB BACK TO FL360; WHICH WAS ACCOMPLISHED. OTHER ACFT PASSED BELOW US AT FL344. AFTER LEVELING AT FL360; ANOTHER VOICE (FEMALE) CAME UP ON MASS; AND WE WERE TOLD TO IDENT IF WE COULD HEAR. WE WERE GIVEN A NEW FREQ; AND WHEN WE SWITCHED; WE WERE TOLD THAT THE MALE CTLR WAS HAVING RADIO PROBS ON THE FIRST MASS FREQ. FEMALE CTLR ASKED US TO RELAY WHAT WE HAD SEEN AND DONE; AND THEN SAID THAT THEY WERE UNABLE TO HEAR US DURING THAT PERIOD. SHE ALSO SAID THAT PARIS HAD PASSED THAT WE WERE CLRED TO FL320; BUT WE HAD HEARD AND READ BACK FL340 WITH NO CONFIRMATION FROM PARIS.