B757-200 FLT CREW REACTS AS TRAINED TO FALSE TCAS RA AT FL350.

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MINDING OUR OWN BUSINESS AT FL350 IN CRUISE WHEN WE SUDDENLY GOT A 'TFC' WARNING. SCREEN SHOWED RED AND WE GOT THE 'CLB' 'CLB NOW' RA WARNINGS AND THE RED TRAPEZOID. I IMMEDIATELY DISCONNECTED THE AUTOPLT AND BEGAN CLBING ABOVE THE RED TRAPEZOID. FO ADVISED ATC AND LOOKED FOR TFC. OUR SCREEN SHOWED ANOTHER AIRPLANE AT FL350. AT APPROX FL356 THE WARNING INCREASED TO 'INCREASE CLB' 'INCREASE CLB NOW.' WE ADDED TOGA PWR AND INCREASED CLB. I ASKED FO OUR SVC CEILING (410) AND INCREASED CLB. TFC ALSO SHOWED FL356. NO MATTER WHERE WE WENT; TFC STAYED. WARNINGS WOULD HAVE ADVISED ME TO KEEP ON CLBING ABOVE THE SVC CEILING OR INTO A STALL. WE ADVISED ATC CLBING TO AVOID RA AND RECEIVED NEW CLRNC TO FL370. THEY SAID NO TFC ANYWHERE. WE OBTAINED A LATERAL DEV AS WELL -- EVENTUALLY WARNING WENT AWAY. AFTER LNDG ZZZ LEARNED FROM THE MECH THAT THE AIRPLANE HAD A LONG HISTORY OF FALSE RA'S. THIS WAS DISTURBING BECAUSE THAT HISTORY WAS NOT KNOWN TO US ON OUR MAINT BRIEF. IT WOULD HAVE BEEN NICE TO KNOW IN THE FEW SECONDS TO MAKE A DECISION WHETHER TO A) INCREASE CLB APCHING STALL LIMIT AT SVC CEILING OR B) STAY IN THE RED TRAPEZOID; WHICH REMAINED AFTER 1500 FT ALT GAIN; WHETHER OR NOT THERE WAS A HISTORY OF FALSE WARNINGS. REFUSED ACFT FOR NEXT LEG WITH RA DEFERRED; EXPLAINED THAT I DID NOT FEEL SAFE TAKING ACFT INTO CROWDED AIRSPACE. USUAL GRIEF FROM MAINT WHO CANNOT UNDERSTAND WHY NOT. FIX IT! NOTIFY THE FLT CREWS OF ACTUAL HISTORY. WE ARE VERY LUCKY WE DID NOT PHYSICALLY HURT SOME OF THE FLT ATTENDANTS WHO WERE STANDING IN THE AISLES WHEN IT HAPPENED.