In general your narrow research area is on a small slice of history (either in theme, time, or location). So I work on Cold War science, mostly in the USA. Pretty narrow compared to the entire sweep of history.

In graduate school, you are required to do some study in somewhat broader fields. So I did "general exams" (which required reading about 300 books or so in one year) in the areas of modern US history, the history of modern physics, the history of modern biology, critical theory, and science, technology, and politics. ("Modern" here means 19th century to the present, more or less.) That's broader than what I do my specific research in, but still relatively focused compared to the entire sweep of history. I took courses that covered somewhat larger bits of history (e.g., I took a course on medieval medicine at one point, and when I was an undergraduate I took courses on "European history" writ large, and "medieval history," and so on), but breadth is always going to be relative.

Now that I'm a professor, I occasionally teach courses that cover a huge swath of history, from the Ancient world to the present. Survey courses like this ("the history of science," writ largely) mean I have to at least get some knowledge on fields I am not really familiar with and not doing research in. So I teach several lectures on Ancient philosophy and technology, for example. Would I consider myself an expert in this area? No way. Do I know enough to give a coherent discussion of this to undergraduates who are being introduced to it for the first time? (And sometimes answer questions that come up on /r/AskHistorians relating to this topic?) Sure.

(For what it is worth, to your specific point, I really did not have a lot of the "periodization" of Western Civ narratives down until I had to teach them. Teaching helps you learn things in a way that reading cannot compete with. Prior to teaching this class, the centuries all sort of slid together. Today my intuitive sense about medieval and early modern history is much better than it used to be, but in graduate school the difference between the 10th and 13th centuries would have been very hard for me to articulate.)

So you can see there are different levels of knowledge here. On my area of research I have very fine-grained knowledge and can give you the ins and outs of the field, the arguments, the new research, what have you. I know what questions have and haven't been asked. On the other areas, I know enough to get by in some situations, but I wouldn't be able to really engage with experts except in a superficial way. If I had dreams of making real contributions there, or having really deep knowledge there, I'd have to do a lot more work.

Sometimes students think that being a historian means having an exceptionally good memory. I assure you, it does not. It does mean that you fill your head with lots of history, though. And as time goes on, that ends up being what you know very well, in the same way that an electrician knows his or her trade through the doing of it. The nice thing about history is that in a sense it can all be cumulative: it's all connected, and the more connections you start to make, the better it is reinforced in your head. On the first pass, when you don't have a preexisting "structure" to hook these things up to, it's very hard to keep track of them and make sense of them. But it gets easier, the more you read.

In my personal experience, no. The human mind can only hold so much. I've had professors who were exceedingly strong in one area who were almost clueless about other historical events beyond a basic understanding.

The best way to describe the study of history is that you start broadly. Learn the beats of western civilization, for example. Then you zero in on an area of interest. The best historians have multiple areas of interest but outside of that they probably wouldn't know more than an average person.