  Stellar tidal streams are sensitive tracers of the properties of the
gravitational potential in which they orbit and detailed observations of their
density structure can be used to place stringent constraints on fluctuations in
the potential caused by, e.g., the expected populations of dark matter subhalos
in the standard cold dark matter paradigm (CDM). Simulations of the evolution
of stellar streams in live $N$-body halos without low-mass dark-matter
subhalos, however, indicate that streams exhibit significant perturbations on
small scales even in the absence of substructure. Here we demonstrate, using
high-resolution $N$-body simulations combined with sophisticated semi-analytic
and simple analytic models, that the mass resolutions of
$10^4$--$10^5\,\rm{M}_{\odot}$ commonly used to perform such simulations cause
spurious stream density variations with a similar magnitude on large scales as
those expected from a CDM-like subhalo population and an order of magnitude
larger on small, yet observable, scales. We estimate that mass resolutions of
$\approx100\,\rm{M}_{\odot}$ ($\approx1\,\rm{M}_{\odot}$) are necessary for
spurious, numerical density variations to be well below the CDM subhalo
expectation on large (small) scales. That streams are sensitive to a
simulation's particle mass down to such small masses indicates that streams are
sensitive to dark matter clustering down to these low masses if a significant
fraction of the dark matter is clustered or concentrated in this way, for
example, in MACHO models with masses of $10$--$100\,\rm{M}_{\odot}$.
