Instructors will go or not go depending on their judgement as to whether the student will get value out of the lesson in the current weather conditions (assuming they aren't so bad as to be below the school's minima).
For example an instructor has taken me up for a lesson on a day with no visible horizon, and cancelled another student (at about the same stage in the course) on the same day, because I didn't have a problem with not having a horizon to look at and the other student did.
On another occasion an instructor offered me the choice: there was lots more cross wind than I'd been up in before, "did I want to go up and have a go, and possibly waste my money if I really couldn't cope with the landings, or would I prefer to cancel"? We went flying. I did two perfectly good landings, somewhat to my surprise ... but frightened the instructor (the only time I've visibly done so) on the third one.
And finally there was the time the weather looked rather dodgy, with what looked like a squall off the end of the runway. "We can go up and have a look if you like." I thought I was doing OK on the approach when the instructor said "I have control" ... but about one and a half seconds later we were blown all over the place (I have no idea how he saw that coming), and I still don't know how he did such a neat landing with gusts and flurries of heavy rain from all directions at once. I wasn't charged for that "lesson". I did however learn that the aircraft was capable of coping with vastly worse conditions than I would ever want to!