CAPT OF MLG ACR ACFT RESPONDED TO A TCASII RA AND CLBED ABOVE ASSIGNED ALT.

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FLT RECEIVED A TCASII RA DURING CLBOUT FROM BRADLEY INTL ARPT. WE WERE HDG 360 DEGS TOWARD CAM VOR WITH A CLRNC TO FL230. ZBW GAVE US A TURN TO 025 DEGS FOR EXPECTED CLB. THEN THE FLT WAS INTERRUPTED BY AN AMBER 'TA' ON TCASII AT 10 O'CLOCK AND AN AURAL ALERT 'TFC -- TFC' AS WE PASSED FL225. THE INTRUDER ACFT WAS TAGGED PLUS 15 WITH A DOWN ARROW. A 'RA' AT 9 O'CLOCK FOLLOWED; TAGGED PLUS 10 WITH A DOWN ARROW. THE AURAL MESSAGE 'CLB; XING CLB -- CLB; XING CLB' AND CLB DISPLAY WERE FOLLOWED THROUGH FL230 AS WE DETERMINED THE INTRUDER ACFT WAS DSNDING THROUGH OUR ALT. A REVERSAL CORRECTION ADVISORY 'DSND; DSND NOW! -- DSND; DSND NOW!' WERE FOLLOWED. OUR ACFT DEVIATED TO FL235 AND THEN RETURNED TO FL230. WHERE WE RECEIVED A 'CLR OF CONFLICT' AS THE INTRUDER LAST TAGGED PLUS 4 AND 1 AND 1/2 MI BEHIND OUR POS. ZBW ADVISED US OF OUR ALTDEV AS WE RPTED OUR COMPLIANCE WITH A TCASII 'RA.' I BELIEVE THE TCASII DATA WE RECEIVED WAS CORRECT. SUPPLEMENTAL INFO FROM ACN 208264: PASSING FL225 THE TCASII PICKED UP TFC AT OUR 10 O'CLOCK 1500 FT ABOVE AND 4 MI. IMMEDIATELY THEREAFTER (3-5 SECONDS) WE GOT A RA FROM THE TCASII STATING 'CLB XING CLB.' THE TFC WAS AT OUR 9 O'CLOCK; 1000 FT; AND 2 MI. WE COMPLIED WITH THE RA TO CONTINUE THE CLB PAST FL230 BELIEVING THE TFC MUST BE DSNDING ON US. I IMMEDIATELY TOLD ZBW THAT WE HAD CLBED ABOVE OUR ALT RESTRICTION. AT THAT POINT; THE TCASII REVERSED ITSELF AND TOLD US TO 'DSND NOW.' AT FL235; WE BEGAN THE DSCNT. ZBW TOLD US OF TFC AT FL240 AFTER WE WERE COMPLYING WITH THE INITIAL RA. WE CAN ONLY ASSUME THE TFC WAS DSNDING TO FL240 CAUSING THE TCASII TO COMPUTE THAT IT WAS DSNDING ON TOP OF US. THE TCASII DOESN'T KNOW IT IS GOING TO STOP AT FL240; OR THAT WE WERE GOING TO STOP AT FL230. WE WOULD NOT HAVE COMPLIED WITH THE RA HAD WE KNOWN OF THE LEVEL TFC AT FL240. CTLRS MUST GIVE PLTS TFC INFO IN ORDER THAT THEY (THE PLTS) CAN GET THE 'BIG PICTURE' TOO!