The cosmic neutrino background (CNB) is a definite prediction of the standard cosmological model and its direct discovery would represent a milestone in cosmology and neutrino physics. In this work, we consider the capture of relic neutrinos on a tritium target as a possible way to detect the CNB, as aimed for by the PTOLEMY project. Crucial parameters for this measurement are the absolute neutrino mass $m_{\nu }$ and the local neutrino number density $n_{\nu }^{\mathrm{loc}}$. Within the $\Lambda \mathrm{CDM}$ model, cosmology provides a stringent upper limit on the sum of neutrino masses of $\sum m_{\nu }<0.12\mathrm{eV}$, with further improvements expected soon from galaxy surveys by DESI and EUCLID. This makes the prospects for a CNB detection and a neutrino mass measurement in the laboratory very difficult. In this context, we consider a set of nonstandard cosmological models that allow for large neutrino masses ($m_{\nu }\sim 1\mathrm{eV}$), potentially in reach of the KATRIN neutrino mass experiment or upcoming neutrinoless double-beta decay searches. We show that the CNB detection prospects could be much higher in some of these models compared to those in $\Lambda \mathrm{CDM}$, and discuss the potential for such a detection to discriminate between cosmological scenarios. Moreover, we provide a simple rule to estimate the required values of energy resolution, exposure, and background rate for a PTOLEMY-like experiment to cover a certain region in the $(m_{\nu },n_{\nu }^{\mathrm{loc}})$ parameter space. Alongside this paper, we publicly release a code to calculate the CNB sensitivity in a given cosmological model.