I can't say what the reality looks like today, but when I was in college 25 years ago (Harvey Mudd), closed-book take-home exams were pretty common (often with a specific time limit: maybe 3/5/8 hours), and I would have been shocked and aghast at the thought of anyone cheating. We took our Honor Code deeply seriously. (I've given my own students take-home open-book exams with "no outside resources" rules regularly in the past, but I've pretty much concluded that isn't viable anymore. But then, we don't have a formal Honor Code here.)

One particularly striking example was a timed, closed-book math exam where the exam paper just listed four problem numbers from the book. We were expected to open the book, copy down those problems without looking at anything else, and then put the book away. Honestly, that one felt like just asking for trouble and now that I'm a professor myself I think that the prof was unreasonably lazy, but it certainly captured the spirit of the thing. (Technically, that wasn't a Harvey Mudd professor but rather a professor at the adjacent graduate university. Maybe that had something to do with it.)