  We study the Brill-Noether theory of curves on K3 surfaces that are Hodge
theoretically associated to cubic fourfolds of discriminant 14. We prove that
any smooth curve in the polarization class has maximal Clifford index and
deduce that a cubic fourfold contains disjoint planes if and only if it admits
a Brill-Noether special associated K3 surface of degree 14. As an application,
the complement of the pfaffian locus, inside the Noether-Lefschetz divisor of
discriminant 14 in the moduli space of cubic fourfolds, is contained in the
irreducible locus of cubic fourfolds containing two disjoint planes.
