I'm old enough to remember when ASCAP or BMI went after the Catholic Church for photocopying music, and collected big-time. I'm afraid non-profit really is unrelated, except for PR value. (Remember the Girl Scouts and Happy Birthday??)
I was in one workshop about a year ago, where the teacher handed out some exercises consisting of the first one-tenth of copyrighted songs; he said this was the legal limit. (Using the work in its entirety is not.) For photocopying copyrighted printed material in a library, it is one-quarter.
The city of Chicago used to send out these blanket letters to anyone who had any at all outstanding parking tickets within the stature of limitations, with the statement "You may be liable to legal action."
My lawyer's comment? "Yeah, and the sun may not rise tomorrow."
Which doesn't help the sleazy feeling this fear-mongering gives us.