LGT DSNDS THROUGH ASSIGNED ALT AND RECEIVES TCASII RA TO CLB. WHEN HE CLBS ABOVE ASSIGNED ALT; HE RECEIVES ANOTHER TCASII RA.

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ATC GAVE CLRNC TO CROSS 30 NE OF APE AT 24000 FT. RESTRICTION WAS PLACED IN FMC AND ALT WINDOWS BY THE CAPT WHO WAS FLYING. I TOLD HIM I WOULD BE OFF THE RADIOS TO GET THE ATIS FOR CMH. I RETURNED BACK TO NORMAL ATC AFTER ABOUT 2 MINS. WE WERE DSNDING THROUGH FL350. AT 34700 FT WE GOT A TCASII RA TO CLB AT 2000 FPM. CAPT AFTER SHORT PAUSE BEGAN CLB AND I TOLD ATC WE WERE RESPONDING TO A TCASII RA. ABOUT THIS TIME AT 35200 FT WE RECEIVED A TCASII RA TO DSND AT 2000 FPM. THE TFC WAS AT ABOUT OUR 10 O'CLOCK POS AT APPROX 3 MI AT APPROX 2200 FT BELOW OUR ALT. DESPITE ME CALLING OUT THE RA OUT LOUD TO DSND; THE CAPT REMAINED LEVEL AT 35000 FT. AT THIS TIME CTR ASKED IF WE HAD RECEIVED THE CLRNC TO MAINTAIN FL350. I RESPONDED 'NO' SINCE I HAD NOT HEARD ANY DIFFERENT CLRNCS AND THE ALT SELECTOR WAS STILL SET AT 24000 FT. THE CAPT THEN TOLD THE CTLR THAT WE HAD RECEIVED A CLRNC TO 'DSND NOW TO FL350' AND HE THOUGHT THE CTLR JUST WANTED US TO START OUR DSCNT OUT OF FL390 AT THAT POINT AND THAT WE STILL HAD THE XING RESTRICTION AT 24000 FT. THE CTLR TOLD US TO MAINTAIN FL350 FT. THE CAPT SHOULD HAVE TOLD ME THE CTLR'S REQUEST WHEN I CAME BACK ON THE RADIOS. EVEN THEN; WE PROBABLY WOULD HAVE DSNDED THROUGH FL350 SINCE THAT IS WHAT HE THOUGHT OUR CLRNC WAS. SINCE THE CLRNC WAS AMBIGUOUS; HE SHOULD HAVE CLARIFIED IT WITH ATC.