B737-700 CREW HAD ALT OVERSHOOT IN ZKC CLASS A AIRSPACE.

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INADVERTENT ALTDEV. NIGHT VMC ON LAKES 4 DEP FROM MCI TO PVD CLRED TO FL290. THEN CLRED TO FL370 WAITING FOR SEPARATION FROM COMPANY ABOVE US AT FL410 BEFORE WE COULD CLB. DURING CLB TO FL370; I CONCLUDED WE COULD GET IN FRONT OF COMPANY WITH FASTER CLB SPD. WE HAD 20 KTS TAILWIND OVERTAKE COMPANY. I DID NOT REALIZE COMPANY WAS ALSO HEADED TO PVD. INADVERTENTLY; I ALLOWED THE AIRSPD TO INCREASE IN THE CLB AND ENGAGED LEVEL CHANGE. THE CLACKER WENT ON. CTR WAS GIVING US VECTORS AND FREQ CHANGE. I SLOWLY PULLED THROTTLE BACK AND IT APPEARS BECAUSE OF ALL THE NOISE WE MISSED THE 1000 FT ALARM. AT FL368; I DISENGAGED THE AUTOPLT; PULLED THROTTLES TO IDLE AND SLOW PUSHED FORWARD. WE LEVELED AT FL378 AND IMMEDIATELY RETURNED TO FL370. CTR ASKED OUR ALT AS WE LEVELED AND WE RESPONDED '370.' THERE WAS NO ADDITIONAL COM. IT IS OBVIOUS THAT I WAS AT FAULT FOR ALLOWING THE AIRSPD INCREASE TO VMO/MMO. THAT COUPLED WITH OTHER TASKS AND NOISE CAUSED US TO EXCEED 370 MOMENTARILY. IN REVIEWING THE AUTOPLT FUNCTIONS; I WOULD RECOMMEND A CAUTION NOTE THAT SAYS THE AUTOPLT MAY NOT LEVEL OFF IF CLBING IN LEVEL CHANGE AT MAX SPD. I ASSUME THIS IS CORRECT SO YOU DO NOT ENDANGER THE ACFT DUE TO BUFFET. TO MY KNOWLEDGE; THERE IS NO MENTION OF THIS IN OUR FOM. SUPPLEMENTAL INFO FROM ACN 540201: THE INCIDENT OCCURRED ABOUT 30 MINS INTO THE FLT. WE EXPERIENCED AN UNINTENTIONAL ALTDEV. THE CONDITIONS WERE NIGHT VMC; CLBING TO FL370 WITH THE AUTOPLT IN USE. PASSING THROUGH FL350; THE MACH SPD WARNING STARTED TO GO OFF (CLACKER). THIS RESULTED IN THE AUTOPLT SWITCHING TO LEVEL CHANGE MODE AUTOMATICALLY. AS A RESULT THE AUTOPLT REACTS BY INCREASING ANGLE OF ATTACK AND RATE OF CLB TO REDUCE AIRSPD. AT THAT POINT; ATC CALLED US WITH A VECTOR FOR SPACING. I BELIEVE THE CAPT REDUCED RATE OF CLB PRIOR TO THE CLACKER GOING OFF BECAUSE OF AN INTERMEDIATE LEVELOFF AT FL370 (I FILED FOR FL410) ATTEMPTING TO GAIN SPACE NEEDED TO EXPEDITE THE CONTINUED CLB TO FL410 RESULTING IN THE INADVERTENT CLACKER. NOW WITH THE CLACKER GOING OFF AND A RADIO CALL FROM ATC; I WAS WAITING TO RESPOND TO THE RADIO CALL UNTIL THE WARNING HAD STOPPED. WITH THIS DISTR; WE BOTH FAILED TO RESPOND TO THE 'ONE TO GO' ALT ALERT WARNING. THE LEVEL CHANGE MODE HAD PULLED THE NOSE UP WHICH INCREASED RATE OF CLB ABOUT 4000 FPM NOW WITH LESS THAN 'ONE TO GO.' PASSING THROUGH FL370 AND STILL CLBING; WE BOTH RESPONDED TO OUR IMPENDING ALTDEV. THE AUTOPLT WAS DISCONNECTED AND THROTTLES WERE RETARDED AND A RAPID RECOVERY BACK TO FL370 WAS EXECUTED. THE WHOLE TRANSITION FROM PASSING THROUGH FL370 WITH A HIGH RATE OF CLB TOOK US UP TO ABOUT FL377. ATC QUESTIONED OUR ALT AS WE LEVELED OFF BACK AT FL370. WE RESPONDED WITH 'FL370' WITH NO FURTHER INQUIRIES INTO THE INCIDENT. I BELIEVE AUTOPLT SWITCHING TO LEVEL CHANGE AND INCREASING THE RATE OF CLB SIGNIFICANTLY WITH 1000 FT TO GO IN AN ATTEMPT TO REDUCE OUR AIRSPD FAILED TO CAPTURE THE ALT HOLD BECAUSE OF THE EXCESSIVE RATE OF CLB SO CLOSE TO OUR LEVELOFF ALT. THE CLACKER GOING OFF; THE ACFT PITCHING UP WITH SWITCHING OVER TO LEVEL CHANGE; ATC'S VECTOR REQUEST FOLLOWED BY A SECOND QUERY PROVIDED ENOUGH DISTR THAT WE WERE UNABLE TO RECOGNIZE THE IMPENDING ALTDEV UNTIL WE WERE JUST PASSING THROUGH FL370. I WOULD LIKE TO KNOW IF THERE ARE LIMITATIONS TO THE AFOS WITH RESPECT VMO/MMO. LEVEL CHANGE AND HOW ALT ACQUIRE/HOLD WILL BE AFFECTED. I WAS ABLE TO FIND THIS IN THE FRM; NOT SAYING IT ISN'T IN THERE. AWARENESS OF THIS PARTICULAR SIT COULD BE USEFUL TO ALL CREWS IN AVOIDING A SIMILAR INCIDENT.