Some people say that Apple really didn't invent anything, they just created a great UI. And the technology was all there for years, it just needed a designer to walk in and create a user experience that didn't suck.
I was there, doing mobile development on the side (we called it "PDA" development back then) and I remember what the existing technology could do. There is no way you could build the iPhone with 2006 technology.
In order to have a good user interface, you need a big screen. To get a big screen, you need a software keyboard that dismisses and doesn't suck. It's hard to imagine now, but just a few years ago the dominant view was that you could never do any serious work on a software keyboard, and there were multiple, billion-dollar business that their whole business depended on it continuing to be impossible to build good software keyboards. Turns out they were wrong, but they weren't wrong that it was impossible with technology that existed at the time. Capacitive touch is directly responsible for 50% of the iPhone user interface as measured by pixel count.
It's hard to imagine this now, too, but putting a real web browser on a phone isn't just a matter of coming up with the idea. As I write this my web browser alone occupies just under 400MB of RAM. Meanwhile the original iPhone shipped with 128MB. Doing web browsing on a 128MB device is a feat of engineering any way you slice it. But even at the time the iPhone was released, let alone years before, many smartphones were shipping with 32/64MB. If you think it was possible to do real web browsing on 32MB years before the iPhone you might be nuts.
Another one of the most revolutionary aspects of the UI was the animations, and pretty much all of them were GPU-accelerated. The presence of a 3D GPU on phones was not exactly new; Nokia had one in their N93 almost a year earlier. But their screen was small, it didn't have to push many pixels, and the thing weighed nearly half a pound. The industry would have gotten there eventually, but you certainly could not ship an iPhone-like product with 2006 technology.
Perhaps the best argument is this RIM employee who summarizes the feeling back in 2007:
All these companies were fighting over what amounts to overgrown PDAs with phones and wireless stacks strapped on. Everyone assumed power density was no where even close to what was needed for general computing, that a full featured browser and heavy duty Internet services were impossible due to bandwidth and latency. Take a look at how our Java expert groups named standards, how people at the time talked about what features smart phones should have, and its clear that no one thought an iPhone was possible. Even Danger, which eventually [led to] Windows Phone 7 and Android, was just working on a better Blackberry...
RIM was even in denial the day after the iPhone was announced with all hands meets claiming all manner of weird things about iPhone: it couldn’t do what they were demonstrating without an insanely power hungry processor, it must have terrible battery life, etc.