Many of us in the tech industry struggle with this question a lot, especially us designers. And I believe it's because, at the core, we love to learn and we love to create, and there are hundreds of different mediums to do this. From learning and implementing iOS apps to experimenting with new interaction design methodologies on mobile, we have the ability at our fingertips, literally.
It All Depends Upon the Person
If I had to give a young junior designer advice on whether to try and master one discipline or expand their knowledge base by studying multiple disciplines, I'd first try and figure out who that person is.
This is so difficult to admit to ourselves, but what you can, and cannot do, does not lie solely in yourself. It's how you were created. Some people learn faster than others. Some are taller, stronger, prone to learning disabilities, and so on. Qualitatively, we are all created equally, but we are not all created equally in our abilities and talents. We each have strengths and weaknesses.
Not only are we created differently, but we've each set up our lives with different priorities and responsibilities. I have a wife and 4 daughters, so that leaves me with little extra time in the evenings to learn or do something entirely new. So, my 6-8 hours at night are not the same 6-8 hours at night for a single woman out of college. It's just how we each have set up our lives. This plays into our stories as well.
But all of this is great! Who wants each person in this world to have the same strength or knowledge or skillset as everyone else? Who wants all people to be in the same relationships, with the same responsibilities? That's just boring.
John Piper's Story of Weakness
I love the story that John Piper (world-renowned pastor, author and entrepreneur) has told about himself and his ability to read. He said that when he was a professor, he just wasn't able to read quickly, so he decided to take his weakness and strengthen that. Since he read slowly, he decided that he wasn't going to read a large number of books, but the books that he did read, he was going to read as deeply as he could. And now he's one of the deepest thinkers of the modern world.
Try it Out
So, if you're wondering whether you should be a jack of all trades or go the other route and excel at a single trade or skill, find out what interests you the most, then try it out. If you end up finding that you're pretty good at one thing, and can't get a grasp on multiple specialties, then maximize your efforts into that one area. If you are having fun with multiple disciplines and see yourself gaining knowledge and skill in each, keep going.
Each person is different. Each situation is different. Don't worry what others are doing. Do what you're good at, and get better. Learn what you love and don't stop.