It's a common mis-conception, but we're not talking about the stage where I'm trying to get an accurate picture of what you'll be like in the job. It's still the very first stage of the interview process, and we're not yet at the stage where we need to decide which of several demonstrably competent programmers we want. We're talking about weeding out people who simply can't write programs at all.
The realistic view of your ability comes later in a second, or even third interview. At that stage we will also talk about system/program/algorithm design skills, ability to work with others, willingness to travel, preferred working hours, and a whole bunch of other stuff to get an idea of how you add value to the company, and how well you will fit with the existing staff and ethos.
But if you can't write something as simple as FizzBuzz in under 15 minutes in a slightly strange environment, you probably won't be able to program well enough here in this company to bother with the other questions. Working here requires more than just producing code in your ideal environment without working with others. I suspect we're not alone in that.
EDIT: I've corrected a few typos. To the person that down-voted me - thanks - you made me read it again and realise that it wasn't as clear as it might've been. I was on my way to a meeting and wrote it in a hurry. Although still not perfect, I think it's better now.
To others who might choose to down-vote it, I'd appreciate any insights as to why. Sometimes I hold unusual, eccentric, or unpopular opinions, and I'm willing to bear the consequences of that, but I'd really appreciate the opportunity to learn what you disagree with. Please, share your thoughts, don't just down-vote without giving people a chance to consider what you think.