Here's what happened:
He was cleared to depart runway 24 using the TETERBORO NINE departure, then to "climb via the SID, expect FL xxx 10 minutes after departure...". The TEB9 departure requires a couple of intermediate level offs before climbing up to your cleared altitude:
His question was the same as mine and yours - what exactly does the controller mean by "higher"? That's not standard phraseology. Higher than what in a minute? Typically something like this would be used when a delay is expected to the final altitude, FL xxx in this case. But since in this case there are some intermediate altitudes, there are essentially two possibilities:
1. I can't clear you to FL xxx right now, but can soon. Since you were already cleared to climb via the SID, climbing to 2000 is fine but I'll have your "higher" altitude in a minute.
2. I want you down at 1500 feet for now, don't climb up to 2000 yet. I can get you "higher" than you are currently in a minute.
Who really knows what ATC wanted? The pilot chose (wisely) to assume the worst and stay at 1500. Fortunately, immediately upon crossing 4.5 DME he received a climb to 11,000, so the issue resolved itself without any further difficulties.
The FAA has recently (April 2014) been implementing new "Climb via the SID" terminology, which in large part is designed to reduce this kind of ambiguous situation. However, it actually caused the confusion this time.
NBAA has a great write-up and slideshow briefing on "Climb via" (and its sibling "Descend via" for STARs) at the following link. It's worth a read if you fly anywhere that you're commonly issued SIDs and STARs.
Notice that subsequent altitude assignments effectively cancel the "climb via" authorization. A case could be made that in this example, that's exactly what happened.
But I think the most important thing to take out of this scenario is what we all learn in Private Pilot training - if you don't understand what ATC wants you to do, don't assume, ask! If the frequency is so busy that you can't get in a word, then use good judgment and take what action is necessary. In this case, the PIC and SIC both decided they would stay at 1500 - in my opinion, the absolute right move.
A long time ago (yes, in a galaxy far, far away too), someone told me that whenever you're trying to decide to doing something that you're not sure about, think how you would sound trying to explain your decision to a jury (or the NTSB for example). "Well, the controller told me 'higher in a minute', so I went ahead and climbed from 1500 to 2000 because I thought he meant I could go higher NOW, but even HIGHER in a minute." Doesn't sound very convincing, does it?
It may very well have been what the controller wanted, but we don't know. Good communication is the key!