B737 FLT CREW CLBS ABOVE ASSIGNED ALT AT FL280 WITH ZAB.

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LEVEL AT FL270; AWAITING CLB CLRNC. WE WERE CLRED TO FL280. THE AUTOPLT WAS NOT ENGAGED AS I WAS HAND FLYING THE AIRPLANE. I (CAPT FLYING) THOUGHT THE ASSIGNED ALT WAS FL290. PASSING FL283; THE FO STATED '1000 FT TO GO' WHEN THE ALT ALERT BUZZED. NEITHER OF US REALIZED THAT WAS THE WARNING THAT WE WERE 300 FT HIGH. I CONTINUED THE CLB UNTIL APCHING FL290; AT WHICH TIME I NOTICED THE ALT ALERT FLASHING. I XCHKED THE ASSIGNED ALT WINDOW (28000 FT); REALIZING THEN THAT I HAD CLBED TO THE WRONG ALT. AT ABOUT THE SAME TIME; ATC CALLED AND SAID OUR ASSIGNED ALT WAS FL280. THE CTLR THEN ASSIGNED US A 30 DEG TURN TO THE L AND RETURN TO FL280. I COMPLIED WITH HIS DIRECTIVE. WHEN WE LEVELED OFF AT FL280; HE CLRED US DIRECT TO EL PASO. I NOTED ON THE TCASII AS WE LEVELED OFF AT FL280 THAT THERE WERE NO TARGETS ON THE TCASII SCREEN; IE; NO TFC WITHIN 15 MI; THUS; NO CONFLICT EXISTED. I ASKED THE CTLR IF I NEEDED TO CALL SOMEONE ABOUT THE ALTDEV. HE SAID IT WAS NOT A PROB. THIS INCIDENT COULD HAVE BEEN AVOIDED HAD I BEEN USING THE AUTOPLT. I WAS MOST FORTUNATE THAT ANOTHER ACFT WAS NOT CLOSE ENOUGH TO CREATE A CONFLICT. I WILL BE HAND-FLYING THE ACFT MUCH LESS IN THE FUTURE.