Wait, what? So the rule isn't that the pronunciation starts with a vowel sound? Instead it's a more murky thing about how easy it rolls off the tongue?
University starts with the 'y' sound, hence "a university." On the other hand, "an unimpressive fact" uses "an" because "unimpressive" starts with the 'u' (vowel) sound.
Doesn't "university" start with the "y" consonant sound?
University has a kind of "y" sound, like the word "yes", which isn't considered a vowel sound.