The problem with that - and I say this as a generalist myself - is that usually you don't know a lot about everything. You think you do, but it turns out you don't. For instance, I've known "a lot" about web development. Now I have a new job that exposes me to domain experts in web development - and it turns out I didn't know jack.

There's nothing wrong at all with being a domain expert. Most generalists - again, I say this as one - aren't as good at everything as they think.

But you really should not strive for this. You will most likely become an expert in some thing anyways, but it usually makes people bad programmers, because they stick to stuff, simply because they know a lot about it. For example it's a very bad thing to know just one programming language and to know just one paradigm. It also causes you to always use old stuff. I guess it's one of the main reasons for everything using C, Java, XML even if there are things they'd actually prefer if they use about them. Same for servers, like Apache, the MySQL or relational databases in general.

You won't learn anything anyway, but I have been disturbed when I realized how foolish it was to stick to things, because I knew how to use them. Learning other things usually means that you gain a new perspective making you better overall, even when it comes to stuff you already knew a lot about.

The problem is also that it hinders you to even know about stuff you'd probably be very good at. IMO that's the biggest problems with this suggestion. Sometimes the thing that what looks hard and complicated to you becomes the thing you can't live without and eventually become an expert in. Maybe the 'sometimes' could even be replaced with 'often'.