B767 AND BE35 HAD LESS THAN LEGAL SPACING ON DEP FROM CAE.

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CLBING ON DEP FROM CAE; ATC SEEMED BUSY AND TENTATIVE ABOUT CLRNCS. AFTER SOME THOUGHT; CAE DEP FINALLY CLRED US TO CLB FROM 5000 FT TO 10000 FT. DURING THE CLB OUT OF 5500 FT; A TARGET APPEARED ON TCASII AT 12 O'CLOCK AND 3 MI. THIS TFC WAS DISPLAYED AS 1000 FT ABOVE US. AT 6000 FT; THE TCASII GAVE A TA. AT THAT TIME; CAE DEP ADVISED US OF THE TFC. BY THE TIME WE SAW THE TFC IT WAS AT OUR ALT AND LESS THAN 1/2 MI. WE MADE A RAPID L TURN TO AVOID A COLLISION. AT THIS SAME TIME CAE DEP GAVE US SEVERAL CLRNCS AND SINCE WE WERE VERY BUSY WE MISUNDERSTOOD OUR ALT CLRNC; SO WE CLBED THROUGH OUR LAST ASSIGNED ALT. THIS WAS AFTER THE NMAC OCCURRED AND WAS NOT PART OF THE PROB WITH THAT TFC. IF ANYTHING WAS LEARNED; IT WAS THAT AS CREW MEMBERS WE NEEDED TO SLOW THE WHOLE PROCESS DOWN AND MAKE SURE WE UNDERSTOOD ATC INSTRUCTIONS PROPERLY. THIS WAS JUST ONE OF THOSE RARE DAYS WHEN THINGS JUST SEEMED TO NEVER GO SMOOTHLY. SUPPLEMENTAL INFO FROM ACN 455117: CLBING OUT WITH DEP CTL ON A VECTOR (PASSING 6000 FT); WE NOTICED POP-UP TFC ON TCASII; CO-ALT; 5 MI; 12 O'CLOCK. I SPOTTED TFC A FEW MOMENTS LATER (CO-ALT TO SLIGHTLY HIGH) TO 'TURN L NOW!' AT THE EXACT SAME TIME; TCASII GAVE US AN RA -- L TURN. I THEN SAW THE LIGHT AIRPLANE BANK TO HIS L AWAY FROM US AS WE DID THE SAME AWAY FROM HIM. I ESTIMATE MISS DISTANCE AS LESS THAN 1 MI AND AT 500 FT VERTLY. NOT UNTIL AFTER PLT ACTION AND RA (TCASII) DID DEP ADVISE US OF VFR TFC ON THE NOSE!