  Critical probes of dark matter come from tests of its elastic scattering with
nuclei. The results are typically assumed to be model-independent, meaning that
the form of the potential need not be specified and that the cross sections on
different nuclear targets can be simply related to the cross section on
nucleons. For point-like spin-independent scattering, the assumed scaling
relation is $\sigma_{\chi A} \propto A^2 \mu_A^2 \sigma_{\chi N}\propto A^4
\sigma_{\chi N}$, where the $A^2$ comes from coherence and the $\mu_A^2\simeq
A^2 m_N^2$ from kinematics for $m_\chi\gg m_A$. Here we calculate where model
independence ends, i.e., where the cross section becomes so large that it
violates its defining assumptions. We show that the assumed scaling relations
generically fail for dark matter-nucleus cross sections $\sigma_{\chi A} \sim
10^{-32}-10^{-27}\;\text{cm}^2$, significantly below the geometric sizes of
nuclei, and well within the regime probed by underground detectors. Last, we
show on theoretical grounds, and in light of existing limits on light
mediators, that point-like dark matter cannot have $\sigma_{\chi
N}\gtrsim10^{-25}\;\text{cm}^2$, above which many claimed constraints originate
from cosmology and astrophysics. The most viable way to have such large cross
sections is composite dark matter, which introduces significant additional
model dependence through the choice of form factor. All prior limits on dark
matter with cross sections $\sigma_{\chi N}>10^{-32}\;\text{cm}^2$ with
$m_\chi\gtrsim 1\;\text{GeV}$ must therefore be re-evaluated and reinterpreted.
