FLC OF AN MLG WHILE CLBING TO A NEWLY ASSIGNED ALT RESPONDED TO A TCASII RA BY DSNDING TO AVOID ONCOMING TFC. WHEN ATC WAS NOTIFIED OF THE ACTION THEY RECLRED RPTR TO A LOWER ALT AND ADVISED THAT THE NEW ALT WAS THE ONE TO WHICH THE RPTR WAS ORIGINALLY CLRED.

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ZLE CLRED US TO FL280 ON FREQ 127.9. WE WERE THEN SWITCHED TO CLE ON FREQ 119.87 AND CHKED IN CLBING TO FL280. AS WE APCHED VWV VOR; I WAS CHKING MY CHART FOR THE OUTBOUND COURSE WHEN TCASII ACTIVATED AS A TA. TCASII HAD TFC APCHING THE VOR FROM THE W AT FL250. I LOOKED AT OUR ALT AND WE WERE CLBING THROUGH FL241 FT AND APPEARED TO BE ON A COLLISION COURSE. I IMMEDIATELY STARTED A DSCNT AND A L TURN AWAY FROM THE TFC. AS I INITIATED THE DSCNT AND TURN THE TA WENT TO AN RA AND 'DSND' WAS ANNOUNCED. WE PASSED ABOUT 4 MI S AND 700 FT BELOW THE TFC. WHEN THE FO NOTIFIED ATC OF OUR ACTIONS; HE SAID HE HAD BEEN TOLD WE WERE ONLY CLRED TO FL240. HE THEN TOLD US TO MAINTAIN FL240. THIS WAS A CLASSIC CASE OF A POTENTIALLY DANGEROUS SIT THAT WAS AVOIDED BECAUSE OF TCASII. ATC WAS NEVER AWARE OF ANY CONFLICT AND WE WERE DISTRACTED IN THE COCKPIT BY OUR NAV DUTIES. HAD TCASII NOT WARNED US; THERE WAS AN EXCELLENT CHANCE FOR A MIDAIR. THIS WAS DEFINITELY AN ATC SCREW-UP AND CONCERNS ME GREATLY. THERE HAS BEEN A GREAT EFFORT BY MY AIRLINE TO EMPHASIZE SAFETY AND STANDARDIZATION IN THE COCKPIT. AS A CHK AIRMAN ON THIS AIRPLANE; I PERSONALLY GAVE A GREAT NUMBER OF EXTRA LINE CHKS IN THE LAST FEW MONTHS TO ENSURE COMPLIANCE. ONE OF THE POINTS WE STRESSED WAS FOR BOTH PLTS TO VERBALLY RESPOND TO ALL ALT ASSIGNMENTS. THE FO AND I HAD DONE THAT WHEN ASSIGNED TO FL280 AND THE FO SAID WE WERE CLBING TO FL280 WHEN HE CHKED IN. HOW; WITH ALL THESE DOUBLE CHKS; WE STILL ALMOST HIT ANOTHER ACFT IS VERY DISTURBING TO ME. WHAT ELSE CAN WE DO?