Living at the Edge: A Critical Look at the Cosmological Neutrino Mass Bound Cosmological neutrino mass bounds are becoming increasingly stringent. The latest limit within the ΛCDM framework from Planck 2018+ACT lensing+DESI is ∑m_ν < 0.072 eV at 95% CL, very close to the minimum possible sum of neutrino masses (∑m_ν > 0.06 eV), hinting at vanishing or even "negative" cosmological neutrino masses. This paper investigates the robustness of these results by checking the role of potential anomalies in Planck CMB and DESI BAO data, comparing frequentist and Bayesian techniques, and investigating deviations from ΛCDM. Profile likelihood analysis shows constraints agreeing at the ~10% level with Bayesian posteriors. The weak preference for negative neutrino masses is mostly present for Planck 18 data, affected by the "lensing anomaly," but disappears when using the new Planck 2020 HiLLiPoP likelihood, leading to significantly weaker constraints. The pull towards negative masses in DESI data stems from the z = 0.7 bin, which is in ~3σ tension with Planck expectations. Removing these outliers and combining with HiLLiPoP relaxes the bound to ∑m_ν < 0.11 eV at 95% CL. The recent preference for dynamical dark energy alleviates this tension and further weakens the bound. The current cosmological limit is very close to the minimal value in normal ordering and already disfavours inverted ordering. Cosmological limits are typically derived within a Bayesian framework, and as such depend on priors. While the dependence may be weak when the likelihood dominates, it has been established that neutrino masses are strongly sensitive to the choice of prior. There is still no hint of a non-zero mass in the posterior probability density, and it has been argued that cosmological data may favor a negative effective neutrino mass. This could be due to a statistical fluctuation or systematic effect, but it could also indicate new phenomena in cosmology. The current situation requires careful analysis of the origin and behavior of the present constraints on neutrino masses from cosmology. The paper investigates data used, statistical methods, and the extent to which results rely on the ΛCDM assumption. It performs an extensive comparison of cosmological neutrino mass bounds from Bayesian and frequentist perspectives. The use of a frequentist framework can lead to new insights into the sensitivity of the bound on the statistical procedure and help address the role of priors in Bayesian limits. The analysis is complementary to recent Bayesian methods that rely on an "effective neutrino mass" to model negative neutrino mass. The paper also addresses the robustness of the bound to the choice of data, comparing DESI, SDLiving at the Edge: A Critical Look at the Cosmological Neutrino Mass Bound Cosmological neutrino mass bounds are becoming increasingly stringent. The latest limit within the ΛCDM framework from Planck 2018+ACT lensing+DESI is ∑m_ν < 0.072 eV at 95% CL, very close to the minimum possible sum of neutrino masses (∑m_ν > 0.06 eV), hinting at vanishing or even "negative" cosmological neutrino masses. This paper investigates the robustness of these results by checking the role of potential anomalies in Planck CMB and DESI BAO data, comparing frequentist and Bayesian techniques, and investigating deviations from ΛCDM. Profile likelihood analysis shows constraints agreeing at the ~10% level with Bayesian posteriors. The weak preference for negative neutrino masses is mostly present for Planck 18 data, affected by the "lensing anomaly," but disappears when using the new Planck 2020 HiLLiPoP likelihood, leading to significantly weaker constraints. The pull towards negative masses in DESI data stems from the z = 0.7 bin, which is in ~3σ tension with Planck expectations. Removing these outliers and combining with HiLLiPoP relaxes the bound to ∑m_ν < 0.11 eV at 95% CL. The recent preference for dynamical dark energy alleviates this tension and further weakens the bound. The current cosmological limit is very close to the minimal value in normal ordering and already disfavours inverted ordering. Cosmological limits are typically derived within a Bayesian framework, and as such depend on priors. While the dependence may be weak when the likelihood dominates, it has been established that neutrino masses are strongly sensitive to the choice of prior. There is still no hint of a non-zero mass in the posterior probability density, and it has been argued that cosmological data may favor a negative effective neutrino mass. This could be due to a statistical fluctuation or systematic effect, but it could also indicate new phenomena in cosmology. The current situation requires careful analysis of the origin and behavior of the present constraints on neutrino masses from cosmology. The paper investigates data used, statistical methods, and the extent to which results rely on the ΛCDM assumption. It performs an extensive comparison of cosmological neutrino mass bounds from Bayesian and frequentist perspectives. The use of a frequentist framework can lead to new insights into the sensitivity of the bound on the statistical procedure and help address the role of priors in Bayesian limits. The analysis is complementary to recent Bayesian methods that rely on an "effective neutrino mass" to model negative neutrino mass. The paper also addresses the robustness of the bound to the choice of data, comparing DESI, SD