Design is just making things line up.
I think it depends on what you're doing. I read a post somewhere (thought it was on Veerle's blog, but I can't find it) that said most designers have a good understanding of one of the Creative Suite apps, but quite limited knowledge of the others.
A lot of web designers probably get by with little more than Photoshop and probably do fine. If you're doing logo design, Illustrator is pretty much a must, as Nora says. Illustrator is also useful for a lot of print work, or even InDesign if you're making something more like a magazine. And then there's the whole world of video...
Obviously knowing more programs is useful. I recently spent several hours longer on a project because I needed to remind myself how to do things in Illustrator, and I know I'll have to devote quite a bit longer to learning more about InDesign soon.
But learning how to use these programs is relatively easy. The difficult part is learning how to design nice things. So if you can already do that, I'd say you're well on the way...