Low-redshift probes, such as baryon acoustic oscillations (BAO) and supernovae Ia luminosity distances, have been shown to be crucial for improving the bounds on the total neutrino mass from cosmological observations, due to their ability to break degeneracies among the different parameters. Here, we expand background observations to include $H(z)$ measurements from cosmic chronometers, distance moduli from gamma ray bursts (GRBs), and angular diameter distances from galaxy clusters. For the first time, using the physically motivated assumption of positive neutrino mass, we find that neutrino mass limits could be at 95% CL below the minimal expectations from neutrino oscillation probes, suggesting possible nonstandard neutrino and/or cosmological scenarios. Interestingly, it is not only the combination of the three background probes that is responsible for the $\ensuremath{\sum}{m}_{\ensuremath{\nu}}<0.06\text{ }\text{ }\mathrm{eV}$ limits, but also each of them independently. The tightest bound we find here is $\ensuremath{\sum}{m}_{\ensuremath{\nu}}<0.043\text{ }\text{ }\mathrm{eV}$ at 95% CL after combining cosmic microwave background Planck data with DESI BAO, supernovae Ia, GRBs, cosmic chronometers, and galaxy clusters, showing a clear tension between neutrino oscillation results and cosmological analyses. In general, removing each one of three background probes still provides a limit $\ensuremath{\sum}{m}_{\ensuremath{\nu}}\ensuremath{\lesssim}0.06\text{ }\text{ }\mathrm{eV}$, reassuring the enormous potential of these low-redshift observations in constraining the neutrino mass.