FLC OF MLG OVERSHOT ASSIGNED ALT OF 3000 DEPARTING OAK.

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AFTER TKOF FROM OAKLAND; FO FLYING; TWR CTLR RPTED 2 MIL ACFT IN THE FLT PATTERN AT ALAMEDA NAS AND INSTRUCTED A FREQ CHANGE TO DEP CTL. THE TCASII GAVE A 'TFC-TFC' WARNING BUT THE HSI SHOWED NO TFC. WE WERE JUST LEVELING AT 2000 FT. AS WE CONTACTED DEP CTL HE CLRED US TO 3000 FT AND ALERTED US TO THE TFC IN THE ALAMEDA PATTERN AS THE TCASII WAS AGAIN ANNOUNCING 'TFC-TFC.' WHILE WATCHING FOR THE ALAMEDA TFC AND LOOKING IN THE DIRECTION CLRED TO TURN I OBSERVED TFC IN THE TURN DIRECTION SOMEWHAT ABOVE OUR ALT. AT THIS TIME THE ALT ALERT WENT OFF INDICATING WE HAD CLBED ABOVE THE SET ALT OF 3000 FT. RECOVERY AND DSCNT WAS INITIATED WITH THE HIGHEST ALT REACHED APPROX 3600 FT. DEP CTL THEN CLRED US TO CONTINUE CLB TO FL230. THE FO (PF) BELIEVED WE HAD BEEN CLRED TO 4000 FT AND THIS MAY HAVE BEEN. THERE WERE NO WARNINGS AND CLRNC INSTRUCTIONS WITH TFC DIRECTION AND ALT NUMBERS AS WELL AS HDG AND ALT CHANGES FOR OUR FLT THE ADDITIONAL CHANGE COULD HAVE BEEN MISSED. THE TCASII; IN THIS CASE; PROVED TO BE GREATLY DISTRACTING WITH ITS LOUD VERBAL WARNINGS AT THE SAME TIME ATC INSTRUCTIONS WERE BEING RECEIVED AND TFC SEPARATION WAS BEING PROVIDED BY ATC. THIS WAS THE CAPTS FIRST FLT WITH TCASII OPERATIONAL (FO'S SECOND). THE REASON NO TFC SHOWED ON THE HSI WAS DUE TO A PUSH BUTTON SWITCH; WHOSE FUNCTION WAS NOT DESCRIBED; THAT WAS NOT ON.