There are true dichotomies that most people nevertheless think are not. The most prominent example is theism vs. atheism. This is a true dichotomy: You are either one of those. If you are not a theist, you are an atheist, by definition. Yet most people think that "theist" and "atheist" are just the extremes of a spectrum, that there's a lot of "in-between" variants. That's not so.
On the other hand, the opposite is also true: There are non-dichotomies that most people think are true dichotomies. Perhaps the most prominent one is the belief in the existence of something. In this particular case there are three options: Either you believe in its existence, or you believe in its non-existence, or you have no belief either way. (This is different from theism vs. atheism because by the very definition of atheism, even someone who doesn't care one way or another is an atheist by definition.)
The prevalence of the idea that belief in something is a true dichotomy is quite prominent. It's extremely common to think that if someone says "I don't believe that God exists" means the same thing as "I believe God does not exist." However, those are two completely different propositions, and are not equivalent!
I like to compare this to the Goldbach's conjecture: The fact that you have no belief in it being true doesn't automatically mean that you believe it's false. There is a third option: You don't know if it's true or not, so you cannot believe either way.
For some reason it can be incredibly difficult to explain to a theist that the fact that someone doesn't believe in God doesn't necessarily mean that he claims that God doesn't exist.