It is suggested that the Minkowski vacuum of quantum field theories of a large number of fields N would be gravitationally unstable due to strong vacuum energy fluctuations unless an N dependent sub-Planckian ultraviolet momentum cutoff is introduced. We estimate this implied cutoff using an effective quantum theory of massless fields that couple to semiclassical gravity and find it (assuming that the cosmological constant vanishes) to be bounded by MPlanck/N 1/4. Our bound can be made consistent with entropy bounds and holography, but does not seem to be equivalent to either, and it relaxes but does not eliminate the implied bound on N inherent in entropy bounds.