This was a seriously painful problem for a while. Took a bit of searching to come up with a very easy solution (which would have been even easier if I hadn’t made a typo).
What was happening:
1. Automatic upgrades of some plugins worked — but VERY slowly (as in it took HOURS). And it would ususally hang at the end, but I could reload the web page and everything seemed fine and worked. Obviously not the slick interface that was intended.
2. Some plugins — including the old WordPress Automatic Upgrade plugin — just failed.