Is it possible to know how to write, but not read?

Sure, one could simply study a chart of letters or symbols, learn how to write them, but never learn what sounds they correspond to. 

anything is possible

Quite often I can't read my own writing. Think I need a pharmacist. 

Sure, here on the internet its easy. Just get a fish and wave it over your keyboard, then your cat will do the writing.

I don't believe so.
You would know how to read, to know what you are writing.

I don't know but each time I come across your question, I pause to think about it. I'll answer this time.
:)

I think it has probably happened at some point.

Hey, this just hit me, like JakobA the unAmerican alluded to, the computer --  maybe one could type like crazy on a keyboard/apparatus but have no idea how to manually write with one's hands.
:)