As you can see, these tests are incredibly powerful even up to third cousins. Theoretically, you should see shared DNA in every first and second cousin and 98% of third cousins. It goes down after this but you even have a pretty good chance to see seventh cousins!
What these tests are less precise about is the actual relationship. So sometimes the test will call someone as a third cousin when they may actually be a fifth cousin and so on. The more distant the relationship, the less precise the relationship assignment. See my article about using the "green chart" and DNA Painter to look at all possible relationships
The number of "expected" cousins is pretty imprecise. It assumes each member of each generation will have 2 - 3 children each. Personally, my mother had 37 - yes 37 - first cousins just on her mother's side! So, each family is different.