I think this is a case of irony being taken literally. I think quite, by definition, means a small amount. But Brits love their irony, so it often gets used to mean a lot. And people, like me, who are trying to work out proper usage contextually interpret it wrong. Plus I think it's generally been adopted to mean both now.
Same with "rather." (She is a rather good singer.)
I could be wrong, but I checked with my Brit resource, and he concurs.
Can you think of any other examples besides quite and rather?