User Roles Don't Stick–Only for Newer Users

1 – I’m helping out the owner of a WP site; this is not my own site
2 – It could be that the WP version (3.5.1) is so old that it’s causing the problem, but it’s so weird that I doubt it.

The issue is that the administrator cannot edit users accounts…but only for some users. And it seems to be newer users. Because I was one of the original authors, my account is really old and it works just fine. The admin made me an admin so I could look into the problem.

But we can’t upgrade newer users who register as members (default) to become authors. And here’s the really weird part…there are no error messages. WP says “user roles changed successfully.” But the changes don’t stick.

Again, older users keep their changes, but newer users cannot. Appreciate any help anyone might offer. Thanks.

===ADDENDUM===ALSO WEIRD===

Somehow, user registration is blocked even though it is allowed in the general registrations page. Also, the Akismet spam-block rate has fallen from about 2k per month to <100 per month as of two months ago, when these issues first arose. Connection…?