Almost all of my everyday bags are designer (a few Kate Spades and a Marc Jacobs at the moment), which is actually kind of unusual in terms of my wardrobe as a whole.
I'd be lying if I said I didn't enjoy the "security blanket" aspect of the branding. It's kind of like insurance that, even if I'm having an off day fashionwise, I've got a chance of coming across as someone with some kind of clue about what I wear and some level of taste.
But I don't think I'm buying these bags because they're a name brand. What happens is usually more like this: I usually want really nice leather and good quality all around because I'm hard on my everyday bags and, well, because nice leather is nice. Then I want something that's current and stylish and also meets my picky functionality requirements. So by then I'm looking for something that's probably not going to be cheap. And, since I'm really deliberative and indecisive on big purchases, I'm going to be most interested in bags I can "visit" a few times in person before I buy, or at least read a zillion reviews of and see a zillion pictures of online. Which, in that range, usually means the big designers instead of something lesser-known.
For things like dressy bags, on the other hand, I'm all about bargain non-designer stuff from wherever I find it. Maybe because functionality and quality are less important for something that only gets used once in a while and I want something cheap enough that I won't feel guilty about replacing it just because I'm sick of it.