Defective DVD

Could be a tiny scratch or a fingerprint or some kind of other smudge on the bottom side of the disc. Take a clean paper towel and some Windex. Spray a little on the disc and wipe from the center of the disc to the rim until the disc is relatively clean.

I used to have this problem with both library DVDs and Netflix DVDs.

It could also be a faulty disc. Small film studios might use inferior DVD discs and the file might become corrupt in the burning process of the film to the disc. But that's highly unlikely unless you're literally the first person to have watched the disc immediately after the library acquistion. The only option here is to return the disc and TELL the library of the faulty disc. Maybe they can return it to the distributor for a better copy.