I've been testing the users in question in our sandbox environment, and log in as users who are reporting the insufficient privileges error by attempting to clone cases created by managers/those with higher level access. Sometimes the error will appear while other times it will not. There's been no consistency in why. It's the same record type.
4 answers
Sounds more like a validation rule to me then, do you see any under
Setup > Customize > Cases > Validation Rules
?
What does the error message look like? Can you provide a screenshot?