LESS THAN STANDARD SEPARATION BETWEEN 2 ACR ACFT JUST AFTER HANDOFF FROM ONE ARTCC TO ANOTHER.

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WHILE CLBING TO OUR ASSIGNED CRUISING ALT OF 37000'; WE WERE GIVEN A TURN TO THE RIGHT TO A HDG OF 100 DEGS AND TOLD TO CLB AT BEST RATE TO FL370. AT THAT TIME WE WERE ABOUT LEAVING FL330 OR FL350; I CAN'T REMEMBER. I HEARD THE CTLR GIVING ANOTHER ACFT A TURN AND DSCNT. I SAW THE OTHER ACFT GO BY US ON OUR LEFT SIDE AND BELOW US. THE OTHER ACFT; ACR Y; WAS ABOUT 4 MI FROM US. WE LEVELED AT 37000' ON A HDG OF 100 DEGS AND THE CTLR CHANGED US TO FREQ 120.55 (ATL CENTER). WE WERE ON HOU CENTER (132.6) WHEN WE TURNED TO THE 100 DEG HDG. WHEN WE CONTACTED ATL CENTER (120.55) THE CTLR SAID WE WERE NOW CLRED LEFT TURN DIRECT TO MGM. I ASKED HER IF THIS CONFLICT HAD SET OFF THE ALARM. SHE SAID YES IT DID SET OFF THE ALARM. I TOLD HER THAT WE HAD BEEN CLRED TO 37000' AND WE WERE USING NORMAL SPDS AND CLB RATES AND WONDERED HOW THIS HAPPENED. SHE SAID THAT THERE WAS A COM PROB BTWN THE SECTORS AND THAT IT WAS THEIR FAULT. THIS HAS HAPPENED TO ME BEFORE A COUPLE OF MONTHS AGO. I WAS HOPING THAT THIS NEW EAST COAST CHANGE IN ROUTES WOULD PREVENT THESE THINGS FROM HAPPENING. SUPPLEMENTAL INFO FROM ACN 84054: I SAW ANOTHER ACFT AT OUR 1-2 O'CLOCK POS AND QUERIED ATL IF THAT WAS OUR TFC. CTLR AFFIRMED THAT IT WAS OUR TFC; AND THAT PLT STATED THAT HE WAS GOING TO TURN LEFT BEHIND US BEFORE TURNING RIGHT TO 090 DEGS; AND I SAW HIM INITIATE THE LEFT TURN. I QUERIED ATL AGAIN AND WAS TOLD THAT THE OTHER ACFT HAD BEEN CLRED FROM FL330 TO FL290. FURTHER QUESTIONING REVEALED THAT HE ALSO WAS LEVEL AT FL310. WE RECEIVED NO OTHER INFO FROM CENTER AS WE PROCEEDED FROM SECTOR TO SECTOR. WE DON'T KNOW WHAT TRANSPIRED BTWN CENTER AND THE OTHER ACFT BEFORE WE CAME UP ON FREQ; NOR DO WE KNOW WHAT THE ACTUAL SEP WAS. HOWEVER; WE DO FEEL THE SITUATION WOULD HAVE BEEN MUCH MORE SERIOUS HAD THE OTHER ACFT NOT FINALLY MADE HIS LEFT TURN BEHIND US.