OK, I have more information, and this is interesting.
First, I'm pretty sure it's not an access issue, and I've also discovered it's not a problem with the link element type. I have a bunch of lists, all set up similar. They generally have a link to a pdf document, with a title, date, etc. This is a government site, listing tax forms and statutes and things like that.
I don't allow any front end editing. To edit the lists, I go in through the back end and click "View Data" on my list. If I click "edit" on an existing item in the list, it works fine; I can edit all the fields and they display as they should. It also allows saving without a problem.
The problem occurs when I click the "add" button to add a new item. The first element in the list of elements is uneditable - meaning I can't add anything to that field. It doesn't seem to matter which type of element. Generally, I have a link as the first element which is why I thought it was related to links. But no matter what element type it is, as long as it's in the first position, I can't edit it. If I change the order of elements, then whatever is the new first element will now be uneditable.
However, if I go ahead and create the new item and save it, it will save OK, but the first element will obviously be empty. But now I can go in and edit the newly created item; all elements will appear OK, and I can add the missing info into the first element and save it.
How's that for fun?
BTW, I'm on Joomla 2.5.9, and as mentioned, Fabrik 3.0.7.