Authors are often fascinated by what would make an agent drop them. For me, it's pretty hard to do and rare that I've ever dropped a client, although it has happened. The most obvious reason is that the client is not writing books I think I can sell. There have been times when I've taken on a client for a book that I loved and may have even sold, but when the next book comes in I just don't think it's right for me. Maybe it's a different genre or maybe I just don't like it. Either way I don't think I could do my best for this client and another agent would probably be better for her.
I've also dropped clients (only a few) for personality reasons. Usually this happens very soon after signing and needs to be extreme. I'm not planning on being best friends with a client, but I do need to feel like we have similar communication styles and I don't need someone who is going to be toxic. I don't need to be yelled at and I don't need people swearing at me. If that's the case I'll be more than happy to let you find another agent who will listen to you.
And lastly, I've dropped the occasional client because I didn't think she could complete the revisions I thought the book needed before I could begin submitting. This was a mistake I made early on in my agenting career. I would ocassionally take on a client with potential and work with her on revisions. I don't do that anymore, and neither do most agents. It's too risky. If the book needs minor revisions, fine. But if I don't feel I can send it out as is and be proud of it (even without the revisions) I won't offer representation. I might, however, reject the book and give revision suggestions.
What I don't do: I don't set a timeline and drop a client if she hasn't sold after six months or even six years. I don't drop clients because they call too much or ask a lot of questions, and I don't drop clients because we're not friends. As long as I love your work and believe in you I will stick by you through thick and thin. I don't give up easily.
So to turn the tables: What would make you drop an agent?