vejktoro:
"can't" is less formal but can replace both cannot and can not, and it can move: "can't you sleep?"

Perhaps your dialect differs to mine, but to me there is a clear difference between cannot and can not.

"Cannot", which can be abbreviated to "can't", means to lack the ability to do something.
"I cannot sleep" means I lack the ability to sleep, I must stay awake.

"Can not" means to have the ability to not do something.
"I can not sleep" means I have the ability to not sleep, I can stay awake.

Have a look at my post on top of page 5 in this thread. I think we are saying the same thing.
Prescriptive grammarians will require the use of 'cannot' to avoid the confusion in writing.

They invented visibly attaching 'not' to 'can' where there is no choice. For example, this person or that simply canNOT sleep - no choice - while writing the two words separately when there is a choice. (Note that you change your speech stress pattern to indicate this)

This is merely an imposed writing convention to separate the two 'NOTs' in learned script. As much as your mid school teacher wishes, it is not universally accepted. If your audience all finished first year English grammar at University, they might be interested enough to retain the artificial difference, but for writers who wish to be clear to every reader, I suggest they step down off their high horses and visibly stress that second 'not' with italics or what not - as they would in any other situation (have to/must/ought/etc...)

And yes, what I am calling the other NOT does not work as an abbreviation.