  We prove two general results concerning spectral sequences of
$\mathbf{FI}$-modules. These results can be used to significantly improve
stable ranges in a large portion of the stability theorems for
$\mathbf{FI}$-modules currently in the literature. We work this out in detail
for the cohomology of configuration spaces where we prove a linear stable range
and the homology of congruence subgroups of general linear groups where we
prove a quadratic stable range. Previously, the best stable ranges known in
these examples were exponential. Up to an additive constant, our work on
congruence subgroups verifies a conjecture of Djament.
