Which of the following sets could NOT be the lengths of the external diagonals of a right rectangular prism [a "box"]? (An [i]external diagonal[/i] is a diagonal of one of the rectangular faces of the box.)

$ \textbf{(A)}\ \{4, 5, 6\} \qquad\textbf{(B)}\ \{4, 5, 7\} \qquad\textbf{(C)}\ \{4, 6, 7\} \qquad\textbf{(D)}\ \{5, 6, 7\} \qquad\textbf{(E)}\ \{5, 7, 8\} $