B737 FLT CREW WITH ZNY CLBS ABOVE ASSIGNED ALT DURING WX AVOIDANCE EFFORTS IN N90 AIRSPACE.

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FLT DEPARTED PHL. WX WAS IFR WITH SMALL TSTMS IN THE AREA. WE DEPARTED RWY 9L AND TURNED W ON A VECTOR TO INTERCEPT THE MXE 278 DEG RADIAL. OUR ALT CLRNC WAS TO 12000 FT. AFTER ROLLING WINGS LEVEL HDG W; PASSING APPROX 9000 FT; WE DETECTED A STORM CELL AT 12 O'CLOCK POS AND 7-8 MI. JUST THEN; PHL DEP HANDED US TO NEW YORK. WE HAD ASKED TO DEVIATE S; BUT WERE TOLD TO ASK NEW YORK FOR THIS. NEW YORK DID NOT ANSWER SEVERAL QUERIES; SO WE STARTED A TURN S AT ABOUT THE TIME WE STARTED BREAKING OUT OF THE RAGGED TOPS AND COULD SEE THE CELL. DUE TO THE TURB AND TRANSITION FROM IMC TO VMC; OUR LEVELOFF AT 12000 FT WAS OVERSHOT AND THE ACFT REACHED 12700 FT BEFORE IT COULD BE RETURNED TO 12000 FT. THERE WAS NO TFC ON THE TCASII AND NEW YORK SAID NOTHING. POOR COMS; BAD WX AND THE NATURAL TENDENCY TO 'SEE AND AVOID' CONTRIBUTED TO THIS SIT. THE ACFT WAS BEING HAND FLOWN TO MAINTAIN PROFICIENCY. THIS SIT WOULD HAVE BEEN PROBABLY MITIGATED THROUGH USE OF THE AUTOFLT SYS; BUT I STILL THINK THAT WITH THE TURB; IT TOO WOULD HAVE OVERSHOT THE ALT.