The grid wires in the accelerator region of neutral beam injectors can be subjected to high heat fluxes from charge-exchange ion impingement, electron impingement, and radiation. Since the actual values for these heat fluxes are not yet well known, it is useful to determine the limits on the allowable heat flux for various grid cooling schemes. The first example chosen for this parametric study is the potentially interesting case of helium-cooled TZM grid tubes. The results indicate that sputtering and thermal deflections are the two most severe limits on the heat flux that can be tolerated by the grids.