For some people, maybe. Although I don't like the term "outgrow," it's implying that someone that moves away from playing video games is suddenly maturing or something. Video games are just another form of entertainment and art, you are never too old for them.
I've sort of moved away from playing them, I don't play them nearly as much now. My interests have expanded, I've been focusing more on music, playing the guitar, and playing the drums in a band, and a lot of music-related stuff for school is coming up. Although that's only partially the reason.
I like to think all the art that I expose myself to has an influence that both opens my mind to new ideas, thoughts, and feelings and helps me grow both emotionally and intellectually. Artistic input influences my output. But lately, a lot of newer video games have been disappointing me in that respect. Sure, for a while, all I basically played was multiplayer shooters with my friends, just bland, repetitive gameplay. And for some reason, despite how with other forms of art my tastes were exactly the opposite, I liked it. The gameplay mechanics, the physics, all that stuff was interesting to me, and it still is. But while interesting and fun, the cold mechanical workings of a video game only go so far without a great story, memorable characters, and a theme and lasting set of emotions that sticks in my head and makes me think. Even games like New Vegas or Mass Effect 2 don't satisfy this. Sure, they have a great story, but holy shit, I am tired of being this sole, manly, heroic protagonist that saves the world. They are the video game equivalent to science-fiction action movies, and there is nothing wrong with that, that shit's great. But it seems like all new video games out there are like this. What about comedy, drama, romance, and the many, many other infinite possibilities out there? And I find there is a severe lack of good themes, lessons, and moral and social issues in a lot of games.