Learn everything about the production, produce your own game and release it.
There are so many things that can go wrong with the production in a game. When everybody knows what the problem is and how to fix it, the work can be divided and solved much quicker.
You will become a better artist or programmer if you learn the entire route to game production. When you are an artist you can learn how to tailor your art towards the mechanics and when you are a programmer you can tailor your mechanics towards the art. I personally feel that the design of games compared to older forms of media such as film, literature and music is much poorer and if everybody learned the entire process it would make the design of games much better.
If you want media with a great design, all of the parts have to easily fit together. When you experience a production backlog because the art doesn't jive with the programming then the design of the game starts to feel disjunct. This could all be solved if everybody knew the entire process of creating games.
There is a difference between talents and skills. You have to know the difference between the two. You want to get hired because of your tallent and not your skill. Talents is an area of production that you are extremely good at and know a lot about. Skills are something that you can just do and get it done. For example, I have coded many games from scratch, but I am not a talented programmer. I have made the art to many of the games I have produced, but I am not a talented artist. My two talents are game design and composition. Surprisingly there is little difference from composing a symphony and designing a game.
Why don't schools teach this? The answer is that most schools want you to end up in a big company that where the work is very specific. While that is a great prospect for students, it doesn't explain to them that you might be on contract before you get hired full time. So what do you do between contracts. Why don't you make your own game and try and sell it.
During the great recession, I noticed that a lot of people who were laid-off started their own studio in their basement. Not every developer made money but some actually made what they were making in their jobs. I tell this to everybody I meet who wants to be apart of the industry especially artists. Start your own studio, you get to learn (probably more than you did in school) some new things and possibly make some money. Plus, you get to add a game to your resume.
People always tell me that it is unrealistic for people to know all the areas of game development. I say that I made Flashes of color all by myself and the game is the size of super mario 3. I made some money with that game and I feel that I could add it to my resume if I wanted to apply to a job. On top of all of that I learned a lot about game production during that game. I always learn something about production during any game.
Tearing down the fabricated walls of game production practices makes games more creative. Again, the best games with the most coherent designs have every part of the game complimenting each other. The best way to do this is that everybody learns all parts to game production.