Yes, really. If you're a highly qualified individual arguing with someone incredibly unqualified, and the unqualified person fails to agree with your expert opinion because he/she has read some random snippet of information on the topic and considers themselves an expert, then a simple "I've studied and trained in this for years, you're an idiot for arguing with me with so little knowledge" isn't really arrogant, is it? Arrogance, by definition, is an offensive display of superiority or self importance. Stating ones credentials and putting someone in their place for having the gall to argue with you with so little knowledge is hardly arrogance. It's merely stating a fact based upon 1) hard work on your part that is entirely unappreciated and taken for granted by the "idiot", and 2) the fact that someone is really stupid enough to think they can argue with one so qualified based upon a small snippet of information. Because believe it or not, there are some really stupid questions.
Now, that's one context. Obviously, if you were talking to someone who was well researched and merely wanted to discuss something with you, and they made a case you couldn't respond with, saying "I'm more qualified than you, piss off, you're an idiot for questioning me" is highly uncalled for and, I concede, arrogant. But putting a real idiot in their place is hardly arrogant.