24-120 would not be wide enough for me on a DX body, but like Brian says, a review of your photos will tell you how much you value the 18-24 range.
Coupled with something like a 10-24 or 12-24 it would be workable, but if you're talking about a 2-lens solution then you could also think about things like 16-85 with a 70-300 which gives you a lot more on the long end.
If you're thinking about FX but not ready to take the plunge, my thinking has always been, don't compromise your current photography with a strategy to make an upgrade at some point it the future. All you have to do is miss one great opportunity, and that whole strategy is a major fail. Maybe that means 24-120 isn't a great lens for you right now, but it's certainly questionable as well why you'd want to invest in a moderately expensive DX lens like the 16-85. Personally I don't put much stock in these mid-range zooms. The ultra-wide and telephoto ranges are much more important to me, so I splurge there, and I'm happy to cover "normal" with a good, cheap prime. Of course, depending on what and how you shoot, YMMV.
What about a used 18-105VR? Something like that gives you some extended range, but without sacrificing the wide end, and if you move to FX you probably resell it with little to no loss.