Why are edits not saved on custom field profile?
In this case the error was due to contacts that were tied to disabled sub-types.

They had created a few sub-types some time ago, added individuals of that type and then disabled the sub-type. 

Once I enabled them, they were able to edit using the custom field profiles.

Since they weren't using them, I created a profile to bulk edit their sub-type and pretty quickly cleaned things up.