FO OF AN ACR CARGO FALCON 20 OVERSHOT ASSIGNED ALT DURING DEP CLB AND NOTICED A SAAB 340 DSNDING THROUGH THEIR FLT PATH. THE FO IMMEDIATELY TURNED AND CLBED IN ORDER TO PROVIDE SEPARATION. CAPT RPTED CLB TO ATC. ATC ADVISED THAT THE SAAB WAS DSNDING IN RESPONSE TO A TCASII RA DUE TO FALCON 20'S CLB AT A HIGH RATE TOWARD THEM.

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I WAS FO ON THIS FLT; AND THE PF. WE HAD JUST DEPARTED FNT AND WERE ON A VECTOR TO INTERCEPT AN AIRWAY. WE WERE ASSIGNED TO CLB TO 11000 FT AND WERE TOLD ABOUT TFC THAT WOULD PASS R TO L; LEVEL AT 12000 FT. AS WE NEARED OUR ALT; ZOB ASKED IF WE WERE ASSIGNED A VECTOR TO INTERCEPT OR ON A HDG. I WAS CLBING AT A HIGH RATE (APPROX 2000- 3000 FPM) AND LOOKED IN TO CHK MY NAV RADIO. I WAS ON A 270 DEG HDG TO INTERCEPT. WHEN MY EYES WENT BACK TO THE PANEL; I SAW US JUST ABOUT AT 11000 FT; WITH THE SAME CLB RATE. THE CAPT; WHO WAS CHKING THE NAV FREQS; CAUGHT IT AT THE SAME TIME. I PROMPTLY PULLED THE PWR BACK AND PUSHED THE NOSE DOWN TO STOP THE CLB. AS I DID THIS; WE NOTICED THE OTHER ACFT APPEARED TO BE DSNDING. WE FELT OUR BEST COURSE WOULD BE TO CONTINUE THE CLB AND TURN BEHIND HIM. I LEVELED AT 11600 FT AND THE CAPT CALLED ZOB TO ANNOUNCE OUR DEV AS WE PASSED ABOUT 1 1/2 MI BEHIND AND 400 FT ABOVE THE OTHER ACFT. AT THE CONCLUSION OF OUR TRIP; THE CAPT CALLED THE HANDLING CTLR TO GIVE FURTHER EXPLANATION IF NEEDED. HE SAID THE AIRLINER'S TCASII PICKED UP OUR HIGH RATE OF CLB AND ISSUED A DSND COMMAND; AND THE PLT BEGAN A DSCNT. THE CAUSE OF THIS WAS MY DISTR TO INSIDE THE COCKPIT WHILE CLBING AT A HIGH RATE; AND APCHING MY ASSIGNED ALT. IF WE HAD NOT SEEN THE OTHER ACFT DSNDING; I WOULD HAVE LEVELED AT 11200 FT AND MAINTAINED 11000 FT. THE CAPT AND I DISCUSSED THE PF AND PNF RESPONSIBILITIES AND HOW THEY WERE NEGLECTED FOR AN INSTANT. I SHOULD HAVE SLOWED MY CLB RATE FOR THE LAST 1000 FT; CONTINUED MY NORMAL SCAN; AND LET THE CAPT HANDLE SETTING THE RADIOS. I ALSO FEEL TCASII IN THE OTHER ACFT PLAYED A ROLE.