ACR B737 RECEIVED A TCASII TA ON ACR LGT CLBING THROUGH HIS ALT. THE TA TURNED TO AN RA AND THE B737 STARTED A CLB AND QUESTIONED THE ARTCC CTLR ABOUT TFC. ARTCC CTLR TURNED THE B737 AND THEN CLBED HIM TO HIGHER ALT. PROX OF TFC INDICATES THAT SEPARATION WAS NOT LOST. ARTCC CTLR HAD NO COMMENT OTHER THAN A FREQ CHANGE WHEN QUERIED BY THE B737.

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WE WERE PROCEEDING AT FL290. WE RECEIVED A TCASII TA FOLLOWED IMMEDIATELY BY AN RA DIRECTING US TO CLB. THE FO INITIATED A CLB. I SAW A TARGET (RED) ON THE TCASII TO THE N OF US CLBING THROUGH FL280; DISTANCE APPROX 5 MI. I ADVISED ATC OF THE SIT AND TOLD ATC WE WERE CLBING. HE GAVE US INSTRUCTIONS TO TURN R TO 090 DEGS. WE HAD BEEN ON A NORTHERLY HDG. I THEN VISUALLY SIGHTED THE APCHING ACFT. IT WAS A B757 OR AIRBUS; I THINK (DEFINITELY NOT CERTAIN OF TYPE). HE WAS HEADING IN A SOUTHERLY DIRECTION AND PASSED ABOUT 5-7 NM TO THE W OF US; CLBING THROUGH FL290. THE CTLR SUBSEQUENTLY CLRED US TO CLB TO FL330. AT THE TIME WE INITIATED OUR CLB THERE WERE NO OTHER TARGETS ON OUR TCASII. WE DID NOT CLB ABOVE FL298 DURING THE MANEUVER; SINCE WE HAD ACQUIRED THE CLBING ACFT VISUALLY. IN HINDSIGHT; WE WOULD NOT HAVE COLLIDED; BUT CLRLY; FOR SOME REASON; THE OTHER ACFT HAD CLBED THROUGH OUR ALT WITH LESS THAN REQUIRED SEPARATION. THE CTLR WAS BUSY AND HAD NO COMMENT ON THE SIT. HE GAVE US A FREQ CHANGE AFTER CLRING US TO FL330; AND WOULD NOT RESPOND TO MY INQUIRY. I DID NOT PRESS HIM BECAUSE HE WAS BUSY WITH SEVERAL OTHER ACFT.