Well, a lot of times a mental institution is just shortened to "The Institution." I've heard of similar experiences to what you're describing happening to escaped mental patients.
Maybe you were imagining what it would be like if you were a paranoid mental patient who felt trapped and wanted to escape?
It's not called "The Institution," but it sounds very similar to Frame of Mind, a Star Trek episode. Except without the part where you go back inside your mind and fight illusions.