First, know thyself. If you don't have talent, or you do but can't stand hunting that race condition for days, or can't obsess on the tone of green that will convey that company's corporate identity better, you will fight an uphill battle for skills that others master much more easily.

That said, I'd say that the general advice goes like this: If you want to work in a corporate environment with lots of managers to report to, specialize as much as you possibly can. That's "knowing everything about nothing" as my adviser in grad school used to say. Companies with a high number of employees tend to prefer specialists. They of course need people with a high range of skills in upper layers of management, but you won't get there first thing you set your foot through the door, it'll take you years.

But if you'd rather work in a start-up environment, or just be your own boss, you need to know everything from the tradeoffs of apache vs nginx to which css tags render differently in Firefox and Webkit.

That's more a lifestyle choice, in my opinion. You can get successful both ways.