ZME REQUESTED B757 CREW TO CLB FROM FL370 TO FL410. DURING CLB; CREW RECEIVED TCASII RA AND TOOK EVASIVE ACTION TO AVOID AN ACR ACFT.

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REQUESTED TURN TO R 10-15 DEGS TO AVOID TSTM CELL. WHILE ON VECTOR; ZME ASKED IF WE COULD CLB TO FL410. WE SAID WE COULD; SO WE COMMENCED SLOW CLB WHILE MODIFYING HDG 10-15 DEGS TO AVOID UNFORESEEN ANVIL HEAD. RECEIVED TA; FOLLOWED QUICKLY BY RA. AUTOPLT DISCONNECTED AND CLB TO GREEN ARC ON TCASII INITIATED. ACR Y AS WELL AS WE RPTED RA. I THINK ATC FORGOT WE WERE SO CLOSE TO ACR Y OR EXPECTED A GREATER THAN 500 FPM RATE OF CLB AND DIDN'T TELL US. SUPPLEMENTAL INFO FROM ACN 519404: WE WERE IN CRUISE FLT FL370 ENRTE FROM LAS-ATL. APCHING RZC VORTAC; ASKED FOR AND RECEIVED CLRNC TO DEVIATE R (S) OF COURSE TO AVOID WX (TSTM BUILDUPS). DURING THE CLB; AN OVERHANG FROM NEARBY TSTM REQUIRED A FURTHER R TURN OF 20-30 DEGS FOR A LATERAL DISTANCE OF ABOUT 1 NM PRIOR TO CONTINUING ON COURSE TO MEM VORTAC. DURING THIS TURN; AND AS BRIEFLY POPPED OUT OF OVERHANG; RECEIVED TFC RA; DISCONNECTED AUTOPLT; AND CLBED AS RA DIRECTED ABOUT 1000 FT. BRIEFLY GAINED SIGHT OF TFC AS IT PROCEEDED SW AFTER EVENT. NO TFC CALLS OF ANY KIND WERE RECEIVED FROM CTR. BELIEVE THE LACK OF PERFORMANCE OF ACFT DURING HIGH ALT CLB COUPLED WITH WX DEVS CAUSED THE TIMING OF THIS TFC PASSAGE TO NOT OCCUR AS ATC HAD PLANNED. A TIME LIMIT FOR THE CLB FROM FL370 TO FL410 AND/OR NOTIFICATION OF POSSIBLE CONFLICTING TFC WOULD HAVE ALLEVIATED THE SIT.