I always try to remember to use "cannot" when applicable as in "I cannot take an umbrella" (because I do not have one).
I also thought that the problem with "I can not take an umbrella" was that it was ambiguous: it could also mean that I could decide not to take an umbrella. Trying to write unambiguously most of the time, I would therefore never use it.
Now I see it claimed that it is in fact not ambiguous, and only carries the second meaning. Usually I would accept such advice at face value and start using "can not" when the second meaning is unambiguously meant, but this comes from a style guide I have other reasons not to respect very much.
Does "can not" have a single unambiguous meaning?
(I have read http://english.stackexchange.com/questions/4510/why-is-cannot-spelled-as-one-word but my question is more about current usage. The length of the answer to that question does confirm my prejudice a bit, though)