As the subject says: attempting to save changes in the team profile page fails when the profile name is blank (which is quite reasonable in contexts where only one team is needed).
The failure is notified with the brief message "Database error". I couldn't find any more details about the error anywhere: nothing is written to the Apache error log, and couldn't find any custom error log nor anything resembling it within my installation. Discovering that the name field was the culprit was just an accident.
Since the field isn't marked as required, and this behaviour doesn't seem to be documented, I'd regard this as a bug / glitch.