User with Custom Role with enough permissions can inline edit some fields that are editable thru inline edit using Administrator or Other roles.
This might be because the fields are not displayed / hidden on the Preferred form for the role.
On saved searches, it will still follow the availability of fields on the Preferred form for the role.
1. Navigate to Setup > Users/Roles > Manage Roles
2. Edit the Custom Role
3. Under Forms tab, check if there is a preferred form for the record on the saved search (e.g. Transaction, Entity)
4. Uncheck the Preferred checkbox for the preferred Form.
1. Navigate to Customization > Forms > Transaction Forms / Entry Forms
2. Edit the Preferred form for the Role
3. Under Screen Fields tab, look for the field that's not editable thru Inline Edit and mark the Show checkbox.