In my country, future mathematicians usually take only mathematics courses (with a bit of physics and computer science thrown in) from the beginning of their undergraduate studies. Some universities are tougher than others, and students end up learning advanced material pretty early. Some of those students burn out, some have great fun and learn a lot of stuff. On the other hand, a respectable part of working mathematicians comes a longer way, learning something else first (physics, engineering, economics,...).
If you decide to work hard, a good rule to avoid burnout is to check your choice at regular intervals: if you get bad marks and/or need large amounts of coffeine you're doing it wrong. Make sure you always have a social network around you.
I personally spent several years concentrated on mathematics. I had a group of fellow students and we discussed what we were learning in espresso bar patios, during aimless walks and long afternoon teas. I learned a lot and enjoyed it. You should find out what fits you.