Oh, it definitely happens. Especially when you have certain multinational corporations with their own browser and far too much market share who claim their online office suite couldn't possibly work perfectly on <insert competitor>'s browser because it isn't as shiny as their browser.

The only other exception I've seen recently is because of genuinely lacking features e.g. WebMidi, or where a really small outfit just doesn't have the time to ensure 100% compatibility with every browser yet, so puts it out there with heavy warnings.

But IME, it is vanishingly rare to find a website that says "Nope, can't do this with firefox" and flat-out refuse to serve you without a very good reason, and when they do, they're idiots if they use the highly-spoofable UA to do that check.

(But I would love to see counter-examples - as I say, I'm sure they exist, but outside the corporate world, I haven't seen it in a while)