A C560 CREW FLYING A SID AT EWR; PREMATURELY STARTED CLB TO AN INTERMEDIATE ALT.

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THE CREW INVOLVED IN THIS ARE BOTH CAPTS AND HOLD TYPE RATINGS IN THE ACFT. I WAS ACTING COPLT DURING THIS SIT. WE WERE UNCERTAIN AS TO WHOSE NAME WAS ACTUALLY LISTED ON THE FLT PLAN BECAUSE BOTH OF US HAD FILED FLT PLANS DUE TO AN EARLIER THAN SCHEDULED DEP AND THEN AN EXTENDED GND HOLD DUE TO WX. WE WERE IN GND HOLD FOR NEARLY 5 HRS AND HAD RECEIVED AN INITIAL CLRNC PRIOR TO GETTING THE GND HOLD AS WELL AS AN UPDATED CLRNC LATER IN THE DAY. AFTER DEPARTING RWY 24 AT TEB AND FLYING THE SID; WE WERE LEVELING AT 3000 FT WHEN ATC COMES ON AND ASKS WHY WE ARE CLBING. AFTER CHKING THE SID AND OUR DME; WE WERE AT 4.5 DME AT THAT TIME AND THAT WAS MY REPLY. ATC REPLIED THAT THAT WAS CORRECT AT THE TIME BUT WE HAD BEEN CLBING FOR THE PREVIOUS 3 MI. APPARENTLY AFTER INITIATING THE R TURN TO 280 DEGS; THE PF CONTINUED THE CLB TO 3000 FT; AND WE DID NOT REALIZE IT UNTIL AFTER ATC CALLED US ON IT. THE BLAME FALLS ON BOTH CREW MEMBERS SINCE NEITHER OF US CAUGHT THE ERROR. WE HAD BRIEFED THE SID PRIOR TO DEP. MY IDEA OF WHAT HAPPENED IS AS FOLLOWS. PRIOR TO TKOF; THE FLT DIRECTOR WAS ARMED WITH THE GAR PITCH ATTITUDE AND THE HDG BUG WAS SET TO RWY HEADING WITH HDG MODE ENGAGED AND ALT PRE-SELECT ARMED. UPON REACHING THE INITIAL 1500 FT ON THE SID; I THEN TURNED THE HDG BUG TO 280 DEGS AS DEPICTED. AT THIS POINT; I ALSO DIALED IN THE NEXT ALT OF 3000 FT IN THE ALT PRE-SELECT. IT IS POSSIBLE THAT THE ALT CAPTURE HAD ENGAGED BUT ALT HOLD HAD NOT; AND WHEN I CHANGED THE PRE-SELECT ALT TO THE NEXT SETTING OF 3000 FT; THE ALT CAPTURE RELEASED AND WENT BACK TO ALT PRE-SELECT. THIS COULD HAVE CAUSED THE PF TO CONTINUE ON WITH THE CLB UTILIZING THE FLT DIRECTOR AND NEITHER OF US CAUGHT WHAT HAD HAPPENED. MENTAL FATIGUE COULD HAVE BEEN AN ISSUE ON MY PART BECAUSE I HAD BEEN SICK FOR THE 2 DAYS PRIOR; BUT BOTH CREW MEMBERS FELT UP TO THE FLT PRIOR TO DEP AND I BELIEVE WE SIMPLY JUST MADE A MISTAKE THAT WE DIDN'T CATCH. ATC DID NOT MENTION ANY TFC CONFLICTS AND OUR TCASII SYS DID NOT INDICATE SUCH.