The effect of a magnetic field on thermal instability in cluster cooling flows, recently considered by Loewenstein, is reexamined. The important finding that a weak ordered field can result in a generic condensational mode is confirmed using Lagrangian techniques appropriate to a local calculation. However, thermal instability seems possible only if the conductivity is well below its Spitzer value, for all nonradial wavenumbers. Wavenumbers not subject to conductive damping are subject to buoyant oscillations. Assuming that instability is present, we extend its physical interpretation. In addition to radially directed magnetic tension suppressing buoyant oscillations, it is shown that lateral magnetic confinement of high thermal pressure regions in the plasma by radial magnetic field lines is at least equally responsible for the suppression of oscillations and the reappearance of local condensational modes. To the extent that magnetic field lines are radially stretched in the course of accretion, this may become the dominant support mechanism against radial oscillations. The general importance of even very modest magnetic fields for destabilizing thermal time scale perturbations (in the absence of conduction) is emphasized.