  A direct discovery of the cosmic neutrino background would bring to a closure
the searches for relic left-over radiation predicted by the Hot Big Bang
cosmology. Recently, the KATRIN experiment put a limit on the local relic
neutrino overdensity with respect to the cosmological predicted average value
at $\eta \lesssim 10^{11}$ [Phys. Rev. Lett. 129, 011806 (2022)]. In this work,
we first examine to what extent such values of $\eta$ are conceivable. We show
that even under cavalier assumptions, a cosmic origin of $\eta \gtrsim 10^4$
seems out of reach (with the caveat of forming bound objects under a new
force,) but find that a hypothetical local source of low-energy neutrinos could
achieve $\eta \sim 10^{11}$. Second, when such values are considered, we point
out that the experimental signature in KATRIN and other neutrino-capture
experiments changes, contrary to what has hitherto been assumed. Our results
are model-independent and maximally accommodating as they only assume the Pauli
exclusion principle. As intermittent physics target in the quest for C$\nu$B
detection, we identify an experimental sensitivity to $\eta \sim 10^4$ for
which conceivable sources exist; to resolve the effect of a degenerate Fermi
gas for such overdensity an energy resolution of 10 meV is required.
