One realization everyone has to come to, whether you're an artist, writer, or even a video game designer, is that you can't specialize in everything.
Artist usually use a similiar technique in all of their work, as well as writers, and video game designers usually focus on doing one thing (Backgrounds, sound effects, character graphics, or creating the game's plot.)
I do believe, however, that you don't have to write the same thing over and over again. Just because you write one hell of a horror novel, doesn't mean you have to become Stephen King and focus your whole life on horror. The biggest thing isn't whether or not you CAN write a certain genre, it's if you have studied the genre enough to write about it. You need a certain amount of smarts when it comes to mystery to write a mystery novel. You need to know enough about space, technology, and what the future may or may not consist of to write a sci-fi novel... But anyone has the ability to research and learn these things.
Right now I'm writing humorous books. The first book I get published I want to be humor, but I do not want to write humor for the rest of my life, in the slightest. I actually have plans on writing a book about the life of prostitution. It will have a dark approach, and I'll explore the evil that is involved in the life style of a prostitute. It will be like Fight Club, when it comes to having a dark, eerie, and slummy feel to it. I'll do my studies, and learn all I can about prostitution and their life, but there is no way I'm writing comedy forever.
So, no, I don't really believe you have to stick with ONE genre... I do believe you shouldn't go over board though. But I also don't believe certain people are just born with amazing talent when it comes to writing a certain genre... The amazing talent is just how good somebody writes, not how good somebody writes a genre. You study the genre and take lessons, then you let your talent in writing do the rest.