There’s a fine difference between “can not” and “cannot.” The first implies the existence of ability and choice. The second doesn’t. Here’s how it works.
I cannot swim. I can’t. I don’t know how. I never learned. (Growing up a hundred miles from any body of water with a mother afraid of water because a childhood friend of hers drowned in a quarry will do that to a person.) Swimming pools aren’t a sufficient impetus. Sorry.
If I knew how to swim, I could rightly say “I can not swim.” I can choose not to. I can, or I can not.
I can not drive. I can choose not to. I know how, and I’m quite good at it, but I can not drive if I don’t feel like it. However, it’s incorrect of me to say “I can’t drive” because I can. On the other hand, if my car’s in the shop, I can rightly say to someone “I can’t drive to the meeting.” I’m not able to. I don’t have a choice. I can’t.
The “can not” construction isn’t common, but it’s not wrong, either. It might be useful at times. Remember, though, that “can’t” means “cannot,” and can’t be used to mean “can not.”
If you can’t, you can’t. Period. There’s no choice in the matter.
Grammargeddon! 
Dear Karen,
I wanted to “like” this post, but I didn’t find the like-button. Then I wanted to share it on G+. Didn’t find a button for it either. So I’m writing this to tell you that this is a great post, to admit (with shame) that I had no idea these are two different things, and that I will share this post to help aducated the masses later.
It’s always a pleasure reading your posts!
Thank you.
Gill
LikeLike
I think it’s funny that Dictionary.com and Merriam-Webster.com define can’t differently. Dictionary.com takes your definition, while MW says that can’t is the contraction of “can not”.
LikeLike
It’s even more amusing because Dictionary.com uses both M-W and American Heritage (I think) as its sources.
Ultimately, it’s a decision everyone has to make personally. I choose to retain the finer differences that others may feel are unnecessary.
LikeLike