I meant to say no non-breaking spaces.
I was (and still am) unclear as to whether this affects many cells or just a few. Since you can use the format painter from a good cell there clearly are at least some good cells.
The reason I suggested some tests (copying from a known good cell, typing an entry, etc.) is to try to narrow down if the problem is with the table cell or the format of the text being placed into it. Since it seems to be the text format, there may be a style involved. Maybe the bad cells are all a particular style, in which case you may be able to identify something that is different between the style that works properly and the style that doesn't. That is a setting that would apply to either the current document, or to all documents based on the template, depending on your choice. In any case, it is pretty close to being a global setting.