It is feasible to assume that you cannot have more than one grant, since getting a grant means that you are paid for a research project and you cannot work on too many projects at once. So answer should be one. Nevertheless there should be some researchers that have more than one grant. How common is that? Is this field related?
At my institution, most researchers have more than one grant. It is extremely common over a wide range of disciplines, at least in the USA.
Often, one is required to include the "level of effort" in each grant proposal; i.e., the percentage of one's time that will be spent on that project. That number is never 100%; more often, it may be 10-50%.
In many fields, the "principal investigator" on the grant may do only a small part of the actual research, with the bulk being done by students and post-docs. This part is very field-dependent. In some fields, full professors at top universities are expected to have very many grants simultaneously, or at least some very large grants that involve a lot more than the full time of a single person.
Here's one example: the National Science Foundation in the US funds academic researchers. Usually the researchers draws some amount of summer salary from the grant (the rest - most of the grant money - goes to student support and travel). But you can be supported by different grants for different fractions of your summer. It's not impossible to have four active grants, each paying 0.5 months of your summer salary.
You would have to check the conditions of the grants to be sure. This answer is based on my experience with some European grants. Especially for "personal" grants that pay your full salary, it is often not allowed to have multiple of such grants simultaneously. In addition it there are often conditions to ensure you are not being paid twice (from different grants) for the same thing.
The situation in Europe is different from the US in the sense that university employees are paid all year round, not only during the teaching semester. So they don't have to source additional income in summer.