B737-300 FLC OVERSHOOT DEP INTERMEDIATE ALT DUE TO RESPONDING TO A TCASII RA AND THE MISUNDERSTANDING THAT THEY WERE CLRED TO A HIGHER ALT TO WHICH THEY WERE GOING.

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DEP CLRNC TO 5000 FT ON 270 DEG HDG; NEXT CLB TO 7000 FT ON A HEADING OF 360 DEGS. ATC CLRED US TO CLB TO 16000 FT AND A R TURN TO 090 DEGS TO JOIN THE SID. I ACKNOWLEDGED THE CLRNC AND THE FO; WHO WAS HAND-FLYING ACKNOWLEDGED THE ALT ALERT WINDOW SETTING. IN THE TURN; CTR CALLED TFC AT 11 O'CLOCK POS AND 11000 FT. WE SAW THE CARAVAN AND RPTED HIM IN SIGHT. ATC THEN TOLD THE CARAVAN THAT WE HAD HIM IN SIGHT AND THAT WE WOULD BE LEVELING AT 10000 FT. AT THAT TIME WE WERE AT 10000 FT AND CLBING 2500 FPM. VERY SHORTLY AFTER THAT WE GOT A TCASII RA TO MAINTAIN VERT SPD. OUR CURRENT RATE OF CLB FELL WITHIN THE GREEN BAND OF THE RA. I CHKED THE TCASII DISPLAY AND THE CARAVAN WAS 200 FT BELOW US AND ABOUT 4 MI AWAY. TCASII CALLED 'CLR OF CONFLICT.' ATC ASKED OUR ALT AND I RESPONDED OUT OF 12000 FT FOR 16000 FT. HE SAID WE WERE ONLY CLRED TO 10000 FT. I TOLD HIM THAT WE RECEIVED AND READ BACK OUR CLRNC TO 16000 FT; TO WHICH HE SAID THAT WE HAD POSSIBLY RESPONDED TO ANOTHER ACFT'S CLRNC. WE ASKED IF THERE WAS A PROB AND HE SAID NO BECAUSE WE HAD THE OTHER ACFT IN SIGHT.