B757 FLC REACT TO TCASII ALERT WHEN CLBING TO A NEW ZJX ISSUED ALT.

Narrative

DURING CLB OUT OF ORLANDO; WE WERE CLRED DIRECT CROSS CITY AND FL280; AS WE CLBED THROUGH APPROX FL260 ZJX STATED ('TO MAKE SURE WE ARE ON THE SAME PAGE; CLRED TO FL290'). WE READ BACK 'FL290.' CAPT SET IT IN OUR ALT SELECT WINDOW AND I VERIFIED IT. AS WE PASSED FL281 OUR TCASII ALERTED US WITH A TA. AS WE LOOKED OUTSIDE FOR THE TFC; WE REALIZED THE CONFLICT ACFT WAS AT FL290. AT ABOUT THE SAME TIME WE RECEIVED A 'DSND' RA; AND AT THAT TIME; CAPT SAID HE HAD THE ACFT AND TURNED OFF THE AUTOPLT; FOLLOWED THE RA AND DSNDED TO FL280. AS ALL OF THIS WAS GOING ON ATC STATED 'ACR X; MAINTAIN FL280.' WE STATED WE WERE CLRED TO FL290; AND WERE FOLLOWING AN RA. AFTER WE WERE CLR OF CONFLICT; ATC CLRED US TO MAINTAIN FL280. WE QUERIED ATC ABOUT THE INCIDENT. THEY SAID IT WOULD BE SORTED OUT AFTER 'THE TAPES WERE PULLED.' OUR ALT SETTING PROCS ARE VERY STANDARD. I AM POSITIVE THAT WE WERE CLRED TO FL290; BUT I FEEL I SHOULD HAVE BEEN MORE SUSPICIOUS BECAUSE OF THE SMALL (1000 FT) CHANGE IN ALT. LOOKING BACK I WOULD HAVE ASKED ATC TO CLARIFY.