B737 AND UNKNOWN ACFT HAD LESS THAN LEGAL SEPARATION.

Narrative

WHILE ENRTE FROM EWR TO SJC WE HAD BEEN AT A CRUISE ALT OF FL310 WITH A REQUEST IN TO THE CTR FOR FL350. THEY WERE UNABLE TO GIVE US FL350 AT THE TIME OF THE REQUEST DUE TO CONFLICTING TFC. 5 MINS OR SO AFTER THE REQUEST FOR FL350; CTR TOLD US TO CLB TO FL330. WE DID SO AFTER ACKNOWLEDGING THE ALT CHANGE. DURING CLB; FOR 2-3 MINS; CTR CAME BACK WITH A CLRNC TO TURN L 30 DEGS. AT THE SAME MOMENT; WE NOTICED AN ACFT POP UP IN OUR TCASII THAT WAS AT OUR 1 O'CLOCK POS; 5 MI AND +500 FT ABOVE US. WE RECEIVED A TCASII ADVISORY ('TFC; TFC'). WE MADE THE HDG ADJUSTMENT; HAD A VISUAL ON THE TFC AND CONTINUED THE CLB. ATC THEN GAVE US A FREQ CHANGE. THE NEXT CTLR GAVE US FL350 AND CLRED US BACK ON COURSE. NOTHING WAS SAID ABOUT THE SIT AND THE FLT CONTINUED NORMALLY FROM THERE ON. I SUSPECT THAT AT THE TIME OF THE ALT CHANGE; THAT WE WERE AT THE EDGE OF A CTLR'S AREA. HE GAVE US THE ALT CHANGE UNAWARE OF CONFLICTING TFC IN THE OPPOSITE DIRECTION. ONCE IT BECAME APPARENT OF A CONFLICT WE WERE GIVEN THE HDG CHANGE. I SUSPECT POOR COM BTWN SECTOR CTLRS RESULTED IN THE CONFLICT.