I can only say how my studies went, things might be different at different universities. In my first 2 years we did a bit of everything, from solid state to biochemistry, passing through (in)organic and analytical chemistry. After that we got more and more choices in course work and everyone started to branch out in different directions, but still with some common courses as well. Then of course the bachelor and master thesis were more specialised, since those were done in two of the research groups. Those two topics could be far apart though.
As a PhD student you have your project, which can be quite specialised. But since you're often collaborating and a lot of research is fairly interdisciplinary, things tend to branch out a bit into other fields as well.
If researchers can roam from chemistry into physics and the other way around, I wouldn't be too worried about missing out on subdisciplines within chemistry.
I don't know about mastering it all, but there are people around that seem to have gotten quite far. I hope most departments have one of those professors you can ask about almost any topic and get a useful answer.