OK, I figured it out. Under softbody -> Solver, I changed the step size min and max way down to min 1, max 10, and error limit of 0.001. That slowed things down a bit, but got rid of the problem. Also, I noticed that this happened mostly when the rope hit the side of the box, so I increased the outer thickness to 0.2. It doesn't put out the odd vertices, but it now doesn't lay directly on the box -- it acts more like a stiff rope. I have an intuitive grasp of why this worked, but if someone reading this could explain it explicitly, I would appreciate it.
