Yes you can have too many pets, but there is no number that should restrict someone. Some people can easily take care of 30 pets, some could only properly care for one, or none.

I could see myself with 6 or 7 dogs. I could handle more but after that point would worry that I could not give them all the individual time and training that I prefer to give my dogs. I want a cat, and I'll probably decide on another once I adopt the first. I can't have any animals that live in cages, it just doesn't work for me. I forget about them, don't give them enough attention, and don't clean their cage often enough. And even though I'd love a rat or a fish, I know I wouldn't care for them properly past the "new pet" excitement stage. Farm animals I could do, but probably not a lot of them. I do want cows, goats and chickens in the future, but would worry about giving proper care to pigs and horses, even though I'd love to own them both.