  We show that for fermion states, measurements of any two finite outcome
particle quantum numbers (e.g.\ spin) are not constrained by a minimum total
uncertainty. We begin by defining uncertainties in terms of the outputs of a
measurement apparatus. This allows us to compare uncertainties between
multi-particle states of distinguishable and indistinguishable particles.
Entropic uncertainty relations are derived for both distinguishable and
indistinguishable particles. We then derive upper bounds on the minimum total
uncertainty for bosons and fermions. These upper bounds apply to any pair of
particle quantum numbers and depend only on the number of particles N and the
number of outcomes n for the quantum numbers. For general N, these upper bounds
necessitate a minimum total uncertainty much lower than that for
distinguishable particles. The fermion upper bound on the minimum total
uncertainty for N an integer multiple of n, is zero. Our results show that
uncertainty limits derived for single particle observables are valid only for
particles that can be effectively distinguished. Outside this range of
validity, the apparent fundamental uncertainty limits can be overcome.
