A perfect example is the default Autodesk "Ladder" family in the specialty equipment category. When you place the family you will always see the railings at the top of the ladder even though they are far above your cut plane:
This has to do with the category of the family... Specific family categories do not allow you to adjust their visibility based on a cut plane. Notice the "(If category permits)" below.
Who has the almighty power to decide which categories "permit" this action? I have no idea... In the example above, the "Specialty Equipment" category does not permit it! But if you are having trouble with your families visibility you may have to change it's category... Which can be kind of lame!