HS800 CREW MISSES TEB 5 SID RESTR OF 2000 FT AND CLBS TO 3000 FT.

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AT XA50 WE DEPARTED TEB. OUR CLRNC WAS VIA THE TEB 5; BREZY; MAINTAIN 3000 FT. IT WAS READ BACK AND VERIFIED. WE TAXIED OUT AND WERE CLRED TO DEPART RWY 24; MAINTAIN 3000 FT. AFTER DEP WE EXECUTED THE DEP PROC. RWY HDG TO 1500 FT R TURN TO 280 DEGS LESS THAN 200 KTS. AT 1500 FT AND 4.2 DME I ASKED TWR TO BE XFERRED TO DEP CTL. THERE WAS A SEVERAL SECOND DELAY TO CHK IN. AT THE TIME I WAS ABLE TO CHK IN 4.7 DME WE WERE AT 1950 FT. MY CHK IN WAS 'AT 2000 FT CLB TO 3000 FT.' NYC RESPONDED WITH 'ROGER TO 3000 FT.' THE SID'S FINAL ALT CHARTED IS 2000 FT BUT WITH TEB TWR'S CLRNC TO 3000 FT WE CONTINUED OUR CLB AND IT WAS VERIFIED BY ATC. I WAS CALLED BACK BY ATC TO 'VERIFY 3000 FT.' RESPONDED 'ROGER 3.' WE WERE AGAIN ASKED TO VERIFY 3000 FT AND TO TURN TO 360 DEG HDG. READ BACK '3000 FT TURN TO 360 DEGS.' AT THIS POINT; NEW YORK DEP ASKED TO CONFIRM MY DEST OF ZZZ. I REPLIED THAT THAT WAS CORRECT. I WAS TOLD TO CALL A PHONE NUMBER UPON LNDG. I RESPONDED 'READY TO COPY.' WE WERE GIVEN A PHONE NUMBER AND TOLD TO CALL DUE TO 'ALT DEV ON THE SID.' AFTER LNDG AT ZZZ AND COMPLETING REQUIRED PAPERWORK; I ESTABLISHED PHONE CONTACT WITH MY CHIEF PLT. THE CHIEF PLT MADE CONTACT WITH THE PHONE NUMBER GIVEN WITH A RETURN NUMBER AND NAME TO RETURN THE CALL. AS OF THIS TIME NO RESPONSE HAS BEEN RECEIVED FROM NEW YORK DEP. SEVERAL CONTRIBUTING PROBS: 1) GIVEN AN ALT CHANGE TO A SID WITHOUT FULL CLARIFICATION OF INTENT OF HOW TO PROCEED PROPERLY. 2) WHEN CHKING IN WITH DEP AT PROPER ALT WITH CLRNC LIMIT WE WERE STILL CLRED TO A HIGHER ALT. 3) PHONE CALL TO NEW YORK DEP NOT RETURNED. 4) IF PHONE NUMBER IS TO A CELL PHONE; WHY WAS IT GIVEN OVER OPEN ATC CHANNEL. 5) IMPLICATION OF A VIOLATION GIVEN OVER OPEN FREQ. FIXES: WHEN ISSUING A SID CHANGE; ATC'S INTENTION SHOULD BE EASILY UNDERSTOOD. IF A POTENTIAL WRONG ALT IS READ; ATC SHOULD QUERY CREW OR CONFIRM ALT. ATC HAS MANY TIMES REQUESTED A RETURN TO ALT BEFORE; WHY NOT IN THIS SITUATION?