I specialise in visual design - mainly just interactive these days with the odd bit of print. Apart from basic actionscripting and some old school lingo I do no coding at all - never needed to and never wanted to. I studied graphic design, typography, fine art, illustration, art and design history and interactive design. All I ever wanted to do was be creative.
I have to agree with JippyMick - trying to be an expert at everything more often than not results in being fairly mediocre right across the board. But for some clients - usually the smaller ones with less of a budget - the jack of all trades is sometimes more appealling. They might not be able to afford to hire a small team of staff (or need to for that matter), so the one man band becomes a necessity.
I suppose it depends ultimately on who you see as your client. If you want to, say, freelance at a big design house then specific, in-depth knowledge and experience are usually king. But, the small business in need of a simple e-commerce site and maybe some clever marketing material would probably feel more comfortable employing a 'do it all' kinda person.
(all sweeping generalisations i know, but you get the idea)