  Guided by evidence coming from a few key examples and attempting to unify
previous work of Chudnovsky, Esnault-Viehweg, Eisenbud-Mazur,
Ein-Lazarsfeld-Smith, Hochster-Huneke and Bocci-Harbourne, Harbourne and Huneke
recently formulated a series of conjectures that relate symbolic and regular
powers of ideals of fat points in ${\bf P}^N$. In this paper we propose another
conjecture along the same lines (Conjecture 3.9), and we verify it and the
conjectures of Harbourne and Huneke for a variety of configurations of points.
