DNA Testing Detail & Clarification

I’m contemplating having a DNA test to help confirm relationships in my family tree. When I review the DNA detail section on a profile and select “Learn more about how you can confirm this relationship with DNA testing,” I’m told Y-DNA and Mt-DNA cannot be used to confirm relationships with my great-grandfathers because we don’t share paternal or maternal ancestors. Additionally, some of them are too distant for AT DNA testing to be of use, apparently. What am I missing? How can I not have a maternal or paternal ancestor in common with one of my grandfathers?