How is it possible for someone to be both a niece and cousin to the same person?
One way this is possible is through inbreeding. A niece is a sibling's child, and cousin is a parent's sibling's child. So if person *A*'s parent and sibling have relations with each other and share a child *B*, *B* is both cousin and sibling to *A*.

Another way this is possible is through adoption, because when someone is adopted, they can be considered the child of both their adoptive and biological parents. If person *A*'s sibling *B* gets adopted by their grandparents,and then has a child *C*, *C* is both cousin and sibling to *A*, since *B* is both *A*'s sibling and parent's sibling.
It is possible for someone to be both a niece and cousin to the same person if their parents are siblings and they have children with their first cousins. For example, if a woman has a brother and her brother has a daughter, the daughter is her niece. If the woman has a child with her brother's son, that child would be both her niece and her first cousin.