  Cosmological neutrino mass bounds are becoming increasingly stringent. The
latest limit within $\Lambda$CDM from Planck 2018+ACT lensing+DESI is $\sum
m_\nu < 0.072\,{\rm eV}$ at 95% CL, very close to the minimum possible sum of
neutrino masses ($\sum m_\nu > 0.06\,{\rm eV}$), hinting at vanishing or even
''negative'' cosmological neutrino masses. In this context, it is urgent to
carefully evaluate the origin of these cosmological constraints. In this paper,
we investigate the robustness of these results in three ways: i) we check the
role of potential anomalies in Planck CMB and DESI BAO data; ii) we compare the
results for frequentist and Bayesian techniques, as very close to physical
boundaries subtleties in the derivation and interpretation of constraints can
arise; iii) we investigate how deviations from $\Lambda$CDM, potentially
alleviating these anomalies, can alter the constraints. From a profile
likelihood analysis, we derive constraints in agreement at the $\sim 10\%$
level with Bayesian posteriors. We find that the weak preference for negative
neutrino masses is mostly present for Planck 18 data, affected by the
well-known `lensing anomaly'. It disappears when the new Planck 2020 HiLLiPoP
is used, leading to significantly weaker constraints. Additionally, the pull
towards negative masses in DESI data stems from the $z=0.7$ bin, which is in
$\sim 3\sigma$ tension with Planck expectations. Without these outliers, and in
combination with HiLLiPoP, the bound relaxes to $\sum m_\nu < 0.11\,{\rm eV}$
at 95% CL. The recent preference for dynamical dark energy alleviates this
tension and further weakens the bound. As we are at the dawn of a neutrino mass
discovery from cosmology, it will be very exciting to see if this trend is
confirmed by future data.
