And again with this thing in which a key point is stipulated: We're talking about the Web.
That incessant stipulation kills me every time. Because:
And if you do, you are probably doing it wrong. Oh, sure, I build stuff that's on the Web. But not "Websites." I build products. Generally information products. Much more often than not, digital products. But they might be the Web (desktop or mobile, in "HTML 5" or not, with any number of plugins like Modernizr or Bootstrap), CSS, JS (with all it's libraries like JQuery which might as well be languages there is so much to know), native apps (for desktops, for mobiles, and each of these use their own language, mostly customized enough to the SDK that it doesn't matter if you knew it before), or maybe it's hybrid (but then you need to know not just a little HTML but also what the hybrid tool, like Titanium, does), or they are printed bits of paper, or stickers, or package designs, or putting the buttons and lights in a specific place on the case.
So, go up to a serious developer you know, and see if he knows how to code, really well, in HTML, CSS, JS, C++, Java, MySQL or...
Oh, and while you are at it, he knows how to run a press, and an injection molding machine, so he can print stickers and make cases also. Right?
You cannot, CANNOT be good at multiple widely different languages. You can, and should, as a UX guy, be slightly versed in the limits of everything, and have a lot of books or links to go look up what you do not know.