The answer is: both.
You should be a t-shaped person.
You should pick one area that you are really good at, and become the goto guy for that skill.
But you should also be competent enough in related areas that you can do non-expert level work.
In a gamedev example, you might be the rendering guy. You know the engine inside and out and you can do all kinds of black magic with it.
But you should also understand the content tool chain well enough to build "developer art". You should be able to run your build server and generate releases. You should (and this is really important) be able to make decent coffee for other members of your team.
How deep and how broad will depend on your own abilities.