I use old thread like that for practicing FMQ, and I need lots of practice, so that's a money saving use for it. If it passes the tug test I might use it in a quilt, but probably for decorative stitches rather than where it is needed to hold things together. I make quilts that are supposed to see a lot of use and washing. In a wall quilt I wouldn't worry about it, as long as it's good thread that isn't going to lint off and cause problems with the machine. Over the years I have not liked the kind of thread that was sold on styrofoam spools. We live in a humid area where thread keeps better than it might in very dry places.
If I were teaching a youngster to sew, I would not set up potential disappointments such as thread that breaks easily. Not only is it frustrating, but it's not uncommon for people to hang on to their very first something they made, and so this thread my wind up in an item that will be pulled out of a drawer and shown to your student's grandchildren someday!