The Harry Potter books are excellent, but there are a couple of aspects that are really nonsensical.
First off: Quidditch. Is it just me, or is this game really stupid? The rules are absurd, and magic or no there's no way anyone would enjoy playing it. I could go into great detail, but if you've read any of the books then I'm sure you already know what I'm talking about; if you haven't, then complaints about quaffles and snitches won't make any sense anyway. (Or, if you really want to read a scathing analysis, check out Quidditch Sucks by jmachinder.)
Secondly, what's the deal with the Ministry of Magic? I'm reading Order of the Phoenix at the moment, and I'm astounded by how much power the Ministry has. They can change school rules on a whim? Hire and fire teachers? Conduct inquisitions and secret trials? Prohibit teachers from saying anything to students that isn't school-related? Ban all clubs and associations? Is there no limit to the Ministry's authority? Within the wizarding world, the Minister of Magic is apparently omnipotent. Maybe this is a particularly British perspective on government, but it rings false to this American. No citizenry, much less a population of wizards, would stand for such fascism.