Nicht negates the word that it precedes. That's except when it comes at the end of a sentence or a clause, in which case, it negates the whole clause. So yes, the position does matter.
To take one of the other examples,
Ich bin nicht müde. I am not tired. I may be happy, or alert, or even annoyed but the one thing is I'm not is "tired."
Ich bin müde nicht. I am tired, not. (This is more of a British construction than an American construction.)
You may be tired, or s/he may be tired, but I am not the one that is "tired."