I've created a Custom Case Field called Value. It's a picklist with two different options users can choose.
If I try to create a Case with this field as an administrator, it works. If I login as a customer and try to fill out this field in the New Case form we have setup, it produces the following error message:
"We couldn't find the record you're trying to access. It may have been deleted by another user, or there may have been a system error. Ask your administrator for help."
If, on the other hand, I try to make a new case as a customer, but do not pick a value for that picklist (if I just leave it at --None--), the case gets created just fine. So it seems like a permissions issue, but I just don't get it. The field is completely editable in the Layout / Field Security settings.