Anecdotally, my own experience with chronic back issues is the reverse, and jives with the explanation I was given. My lowest 3 disks are degenerating, with the bottom disk almost gone. About 8 years ago I had to have a laminectomy due to a disk bulge pushing against a nerve, and no amount of NSAIDs nor physical therapy would fix it.
I had herniated disks before and after, but only one required surgery. For as long as I can remember, I'll get severe back pain once or twice a year, which eventually goes away. The explanation I received for these events is that it is likely that the outer sheath of a disk is getting torn, stretched, or some other type of trauma. This hurts in and of itself, however, additionally the disk material may begin to bulge. Once bulging, that part of the disk loses access to nutrients and moisture and eventually dries up, with the bulge receding and the damage scarring over. The doctor described this like a grape to a raisin, though I'm sure it looks nothing like the analogy.
Summary, with time, the disk sheathing trauma scars and the bulge dessicates, possibly about the same time NSAIDs are declared of no help.
For me, I definitely notice much less pressure in my back when on an NSAID, and I sleep much better as well.