Under parallel electric and magnetic fields, electrohydrodynamic instabilities in nematic liquid crystals having negative dielectric anisotropy are predicted to exhibit a strong dependence of roll spacing on applied magnetic field. We report studies of the parallel magnetic field dependence of the threshold electric field for instabilities and the roll wavelength at threshold. The threshold electric field exhibits fascinating re-entrant behaviour not predicted by theory. We describe qualitatively new mechanisms for the creation and destruction of the convecting state at higher magnetic fields.