AN A319 CREW; DSNDING INTO LGA; OVERSHOT THEIR ASSIGNED ALT.

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I WAS FO OF A FLT ENRTE BOS-LGA AT 16000 FT MSL. WE HAD BEEN GIVEN MULTIPLE ALTIMETER CHANGES AS WE PROGRESSED TOWARDS A VIGOROUS LOW PRESSURE SYS AND; AS A RESULT; OUR INDICATED ALT WAS 15850 FT MSL. IN ORDER TO CORRECT BACK TO OUR ASSIGNED ALT I SELECTED 200 FPM VERT SPD FOR A GENTLE CORRECTION. IMMEDIATELY THEREAFTER I POINTED OUT TO THE CAPT THAT THE ILS IDENTIFIER DID NOT AGREE WITH THE APCH WE HAD BRIEFED AND HE PROCEEDED TO GET THE NEW ATIS; WHILE I PREPARED FOR THE NEW APCH BRIEFING. 2 MINS ELAPSED WHEN THE ALT ALERTER SOUNDED BECAUSE WE HAD; FOR SOME REASON; CLBED THROUGH OUR SELECTED/ASSIGNED ALT OF 16000 FT TO 16250 FT. I PUSHED THE LEVELOFF KNOB AND SELECTED A VERT SPD TO TAKE US BACK DOWN TO 16000 FT. SHORTLY THEREAFTER ZBW ASSIGNED US THE CUSTOMARY 14000 FT AS WE WERE GETTING CLOSE TO THE DEST. NO MENTION WAS MADE OF OUR EXCURSION TO 16250 FT AND TFC WAS NOTED IN OUR AREA ON TCASII. I HAVE NO IDEA WHY THE ACFT FAILED TO LEVEL OFF AT THE SELECTED ALT. I DO NOT RECALL LOOKING FOR THE BLUE 'ALT' CAPTURE ARMED INDICATION WHEN I SELECTED THE CLB. PREPARING FOR THE NEW APCH WHILE CLBING WAS CERTAINLY NOT A GOOD IDEA AND; I WILL BE MONITORING CLBS/DSCNTS MORE CLOSELY IN THE FUTURE.