A B737-300 CREW; DEPARTING BOS; OVERSHOT ASSIGNED ALT DUE TO MISINTERPING THE CLRNC.

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CLBING OUT ON THE COASTAL 2 DEP. CLRED DIRECT TO YODER INTXN. ZBW ON FREQ CLRED US TO CLB TO FL220 AS WE WERE PASSING 12000 FT. IT WAS THE FO'S LEG AND CAPT WAS ON THE RADIO. CAPT READ BACK CLBING TO FL220. WE WERE THEN INSTRUCTED TO CHANGE TO ANOTHER FREQ. UPON CONTACT WITH THE NEW ZBW CTLR; WE CHKED IN PASSING 17000 FT CLBING TO FL220. THE CTLR CLRED US DIRECT TO CCC. AS WE WERE PASSING FL205 HE IMMEDIATELY BECAME ALARMED AND TOLD US WE WERE SUPPOSED TO LEVEL AT FL200 AND TO IMMEDIATELY DSND BACK DOWN. AT THIS POINT; 3 THINGS HAPPENED SIMULTANEOUSLY: 1) WE WERE AT FL213 AND STARTED AN IMMEDIATE DSCNT. 2) WE SAW TFC AT 11 O'CLOCK; ANOTHER COMPANY LEVEL FLT AT FL210 OPPOSITE DIRECTION AT APPROX 8 MI. 3) WE GOT AN RA ON TCASII DIRECTING US TO CLB. AS WE PASSED BACK THROUGH FL210 DSNDING; WE OBSERVED THE OTHER COMPANY TFC CLOSE TO APPROX 2 MI CO-ALT. WE PASSED AS WE LEVELED AT FL200 AND 1 MI. WE INFORMED ZBW WE HAD THE TFC N SIGHT AT ALL TIMES. WX MAY HAVE BEEN A FACTOR DUE TO BROKEN CLOUD DECKS AND WINDS CREATING TURB AND AN UPDRAFT THAT INCREASED OUR VERT VELOCITY TO 6000 FPM AT ONE POINT PASSING FL180. I READ BACK OUR CLRNC TO FL220 TO 2 SEPARATE CTLRS. I DON'T KNOW WHAT ELSE I COULD HAVE DONE TO CONFIRM IT. BOTH CREW MEMBERS HEARD THE SAME CLRNC. I SINCERELY DON'T REALLY BELIEVE; HOWEVER; THE WX CONTRIBUTED AT ALL; BUT I MENTION IT HERE TO SET THE SCENARIO LEADING UP TO THE EVENT. I FEEL THE CTLR MADE A JUDGEMENT ERROR DIRECTING US BACK THROUGH THE CONFLICTING ACFT'S ALT INSTEAD OF ALLOWING US TO STAY DECONFLICTED BY CLBING UP TO FL220. HIS CALL TO DSND EXACERBATED THE CONFLICT. IT WAS GOOD WE HAD THE TFC IN SIGHT. SUPPLEMENTAL INFO FROM ACN 541626: I DON'T REALLY SEE HOW WE COULD HAVE PREVENTED THIS EVENT FROM HAPPENING. WE BOTH HEARD THE SAME ALT CLRNC AND THE CAPT READ BACK THE ALT TO 2 DIFFERENT CTLRS. SUPPLEMENTAL INFO FROM ACN 541463: AN ACR X CLBING WAS CLRED TO FL200 AND CLBED TO FL220. I REALIZED WE HAD A PROB WHEN I NOTICED THE MODE C READ FL203 AND CLBING. I HAD THE ACR X DSND IMMEDIATELY PRIOR TO THE CONFLICT ALERT GOING OFF. THE NEBOUND BOSTON ARR WAS AN ACR Y B737 AT FL210. HE QUESTIONED WHO THE ACFT WAS AND I ISSUED BOTH ACFT TFC. BOTH ACFT HAD EACH OTHER IN SIGHT. THE ACR X ACFT DSNDED TO FL200 AND CONTINUED ON TO ITS DEST. THE ACR Y ALSO CONTINUED ON TO ITS DEST.