CHALLENGER CL601 FLT CREW HAS AN ALT DEV DURING THE TEB 5 DEP.

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WE WERE TOLD TO TURN TO 220 DEGS AND CLB TO 6000 FT. I WAS THE PF AND MISSED PART OF THE RADIO CALL FROM CTL AS I WAS TURNING AND CLBING AND LOOKING FOR TFC. MY FO READ BACK TO THE CTLR CLB TO FL220 AND TURNED THE ALTITUDE PRE SELECT TO 22000 FT. I THOUGHT WE HAD BEEN CLRED TO THAT ALT AND STARTED A RAPID CLB TO GET OUT OF THE ICING CONDITIONS AS SOON AS POSSIBLE. AS WE WERE CLBING THROUGH 6000 FT THE CTLR CALLED AND ASKED US WHAT ALT WE WERE CLBING TO AND THE FO SAID FL220. THE CTLR ASKED US WHERE WE GOT THAT CLRNC AND HE REPLIED THAT IS WHAT HE HEARD AND THAT IS WHAT HE READ BACK WITH NO NEGATIVE REPLY SO WE CLBED. THE CTLR AT THAT POINT TOLD US TO LEVEL AT 6000 FT BUT I WAS ALREADY AT 6200 FT AND CLBING. I IMMEDIATELY LEVELED OFF AND STARTED A DSCNT BACK TO 6000 FT. AS FAR AS WE KNOW THERE WAS NO CONFLICT WITH OTHER TFC AND THE HIGHEST I WAS WHEN LEVELING OFF WAS 6400 FT BEFORE STARTING DESCENT. THIS WAS A TOTAL FAILURE TO COM AND TO QUESTION A CLRNC LIKE NONE OTHER WE HAD EVER RECEIVED OUT OF TEB. LEARNED THAT WHEN THINGS ARE NOT AS THEY NORMALLY ARE QUESTION AND CHECK BEFORE PROCEEDING.