ALTDEV ALT XING RESTR NOT MET. ALT EXCURSION.

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IN SUM: TKOF FROM RWY 12R SJC RWY HDG CLBING TO 12000 FT. THE CAPT WAS FLYING. ATC-SJC DEP CTL; CLRED US TO TURN R DIRECT TO SJC VOR TO CROSS IT AT OR ABOVE 12000 FT; CLB AND MAINTAIN FL230 FOR THE REMAINDER OF THE DANVILLE DEP RED BLUFF TRANSITION. I READ BACK THIS CLRNC AS STATED. THE CAPT AND I UNDERSTOOD THIS TO MEAN DELETE THE RESTR TO CROSS SJC VOR AT 12000 FT. JUST PRIOR TO XING THE SJC VOR; CLBING THROUGH 14 KTS; ATC; NEXT FREQ ZOA; ASKED US WHAT ALT WE WERE CLBING TO. I TOLD HIM WE WERE CLRED TO FL230. ATC ASKED IF WE WERE GIVEN THE RESTR TO CROSS SJC VOR AT 12000 FT. I TOLD HIM WE UNDERSTOOD SJC AT OR ABOVE 12000 FT. THE ATC RESPONDED TO CONTINUE CLB TO FL230. THE CAPT ASKED ATC IF THERE WAS A CONFLICT OR A PROB ATC REPLIED NEGATIVE; THERE WAS NO CONFLICT OR PROB. SUPPLEMENTAL INFO FROM ACN 304124: APPROX 7500 FT ON CLBOUT; DEP CTL GAVE CLRNC TO MAKE A R TURN TO SJC VOR; AT 12000 FT; THEN CLB TO 23000 FT. FO READ BACK L TURN; AT OR ABOVE 12000 FT. ATC AND CAPT WERE BOTH CORRECT TURN DIRECTION; BOTH MISSED AT OR ABOVE READBACK. FO AGAIN QUESTIONED DIRECTION OF TURN. CAPT REPLIES R TURN. APCHING THE SJC VOR (ABOUT 7 DME) CAPT QUESTIONS THE 12000 FT OR ABOVE XING RESTR. FO REPLIES 'YES; 12000 FT OR ABOVE AT SJC TO CLB TO 23000 FT. CAPT WAS HAND FLYING -- SHOULD HAVE HAD AUTOPLT ENGAGED TO MORE CLOSELY MONITOR CLRNC. CAPT SHOULD ALSO HAVE HAD ATC VERIFY THE CLB CLRNC; NOT JUST ASKING THE FO.