Some point was lunchtime today. I noted that I didn't have new screws, but he thought the old ones would be fine. We set it up, an inch or so above where it had been before, and five minutes later, it was attached to the wall again.
Until we gave it a quick tug to make sure it was secure, and it came back out of the wall.
Oid's determination is that I need drywall screws. Neither of us is sure how it managed to stay in so long before when it didn't last now. It's been there, in active use, for at least the entire three years I've been in the apartment.
So, a trip to Home Depot is on the To Do list for sometime next week, unless they happen to sell drywall screws at the grocery store. Figures, that something I thought would be straightforward ends up having complications.