Do most mathematicians know most topics in mathematics?

How many topics outside of his or her specialization is an average mathematician familiar with?

For example, does an average group theorist know enough of partial differential equations to pass a test in a graduate-level PDE course?

Also, what are the "must-know" topics for any aspiring mathematician? Why?

As a graduate student, should I focus more on breadth (choosing a wide range of classes that are relatively pair-wise unrelated, e.g., group theory and PDEs) or depth (e.g., measure theory and functional analysis)?

Your question is philosophical rather than mathematical.

A colleague of mine told me the following metaphor / illustration once when I was a bachelor student and he did his PhD. And since now some years have passed I can relate.

It is hard to write it. Think about drawing a huge circle in the air, zooming in, and then drawing a huge circle again.

This is all knowledge:

[--------------------------------------------]

All knowledge contains a lot, and math is only a tiny part in it - marked with the cross:

[---------------------------------------x----]
                                        |
Zooming in:
[xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx]

Mathematical research is divided into many topics. Algebra, number theory, and many others, but also numerical mathematics. That is this tiny part here:

[xxxxxxxxxxxxxxxxxxxoxxxxxxxxxxxxxxxxxxxxxxxx]
                    |                    
Zooming in:
[oooooooooooooooooooooooooooooooooooooooooooo]

Numerical Math is divided into several topics as well, like ODE numerics, optimisation etc. And one of them is FEM-Theory for PDEs.

[oooooooooooooooooooρoooooooooooooooooooooooo]
                    |                    

And that is the part of knowledge, where I feel comfortable saying "I know a bit more than most other people in the world".
Now after some years, I would extend that illustration one more step: My knowledge in that part rather looks like

[   ρ    ρρ  ρ         ρ   ρ          ρ     ρ]

I still only know "a bit" about it, most of it I don't know, and most of what I had learned is already forgotten.

(Actually FEM-Theory is still a huge topic, that contains e.g. different kinds of PDEs [elliptic, parabolic, hyperbolic, other]. So you could do the "zooming" several times more.)

Another small wisdom is: Someone who finished school thinks he knows everything. Once he gained his masters degree, he knows that he knows nothing. And after the PhD he knows that everyone around him knows nothing as well.

Asking about your focus: IMO use the first few years to explore topics in math to find out what you like. Then go deeper - if you found what you like.

Are there "must know" topics? There are basics that you learn in the first few terms. Without them it is hard to "speak" and "do" math. You will learn the tools that you need to dig deeper. After that feel free to enjoy math :)
If your research focus is for example on PDE numerics (as mine is) but you also like pure math - go ahead and take a lecture. Will it help you? Maybe, maybe not. But for sure you had fun gaining knowledge, and that is what counts.

Don't think too much about what lectures to attend. Everything will turn out all right. I think most mathematicians will agree with that statement.

The answer to your question is easy:
No, an average mathematician specialized in, say, algebraic geometry could not pass without preparation a graduate level exam on partial differential equations.
Wait, it's worse than that: he couldn't even pass an undergraduate level exam on partial differential equations.
Wait, it's even worse: he couldn't pass an exam in algebraic geometry on a different specialized topic from his own. For example an elementary exam on the classification of singularities if he is specialised in Hilbert schemes.
Conversely I would be very surprised if a notorious analyst who recently got a Fields medal could solve the exercises in, say, Chapter 5 of Fulton's Algebraic Curves, the standard introduction to undergraduate algebraic geometry.

Some remarks
1) What I wrote is easy to confirm in private but impossible to prove in public:
I can't very well write that in a recent conversation XXX, a respected probabilist, abundantly proved that he had no idea what the fundamental group of the circle is.

2) If author YYY wrote an article on partial differential equations using techniques from amenable group, this doesn't imply that other specialists in his field know any group theory.
It doesn't even prove that YYY knew much about group theory: he may have realised that group theory was involved in his research and interviewed a group theorist who would have told him about amenable groups.

3) On the bright side some very exceptional mathematicians seem to know a lot about nearly every subject in mathematics: Atiyah, Deligne, Serre, Tao come to mind.
My sad conjecture is that their number is a function tending to zero as time passes.
And although I couldn't ace an analysis exam, I'm aware what this means for an $\mathbb N$-valued function...

My two cents: unless you have a magical brain, or are some sort of epoch-making genius, you're probably going to find that you can only hold only so much mathematics in your mind at any given time. So, for practical reasons—both with respect to writing a dissertation, and with respect to making a career for one's self—you should probably stick to one or two closely related areas, so that you might have sufficient expertise to make yourself useful to a research institution or to whatever it is that you wish to do with your future.

That being said, I've found that elbow grease and skill in mathematics are often woefully uncorrelated with one another. Rather, skill is often dependent more on how much mathematics one has seen. To that end, I would say, though you should definitely pick a subject area or two to call your own, you should strive keep an open mind and maintain an active interest in as wide a variety of mathematical disciplines as possible.

I often find that reading (even if only casually) about forms of mathematics unrelated to my research areas provides a wealth of new ideas and insights. The more patterns and phenomena you are acquainted with, the better the chance that you'll notice something of interest intruding upon your work, and that might give you some intuition you might not have otherwise had. At the very least, it will help you know what topics or sources (or collaborators...) to look up when you stumble across something outside of your area of greatest expertise.

Edit: One more thing. Linear algebra. To paraphrase Benedict Gross, there's no such thing as knowing too much Linear algebra. It's freakin' everywhere.