ACR MLG ALTDEV OVERSHOT CLRNC ALT. RPTR SAYS DISTR BY TCASII TA.

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DEPARTING RWY 1L AT SFO. CLRNC TO 15000; AT 3 DME; ATC ASSIGNED A 230 DEG HDG. IN THE L TURN AT ABOUT 2600 MSL; TCASII TA SOUNDED. AT ABOUT 3000 MSL; TCASII GAVE THE COMMAND TO DSND; WHICH WE COMPLIED. ONCE CLR OF TFC; WE CONTINUED OUR CLB TO 15000; WITH A 2000 FPM CLB. ATC THEN CALLED TFC AND ASSIGNED 4000 FOR TFC. THIS WAS ABOUT 3600 MSL WHEN ASSIGNED. APCHING 4000 MSL THE TCASII AGAIN SOUNDED WITH A TA (3 TARGETS) WERE ON THE TCASII SCREEN. MY ATTN WAS DIVERTED FROM THE ASSIGNED ALT OF 4000 MSL; BY THE TCASII WARNING; AND AN OVERSHOOT OF 280 FT OCCURRED ABOVE THE ASSIGNED 4000 ALT. THE TCASII SEEMS TO BE A GOOD SYS. ON NUMEROUS TIMES THE TCASII SYS SOUNDED AT THE MOST CRITICAL PORTION OF THE FLT (I.E.; APCHING RWYS WITH PARALLEL RWYS; AT 500 FT TA WITH COMMANDS TO CLB OR DSND; ON THE SNA (19R) DEEP CUTBACK (NOISE ABATEMENT) ON DEP. LESS THAN 1000 FT; IN ATA WHERE VISUAL CONTACT IS MADE; FOR VISUAL SEPARATION; ETC...). THE WARNINGS ARE VERY DISTRACTING AT THE CRITICAL PORTION OF FLT WHERE COMPLIANCE IS REQUIRED. PLT EXPERIENCE HAS TO LOCATE TARGETS; AND TAKE THE BEST ACTION; WITH THE LEAST AMOUNT OF POSSIBLE VIOLATIONS; WITH SAFETY BEING FIRST.