The reliability of quiet-Sun magnetic field diagnostics based on the Fe I lines at 6302 Å has been questioned by recent work. Here we present the results of a thorough study of high-resolution multiline observations taken with the new spectropolarimeter SPINOR, comprising the 5250 and 6302 Å spectral domains. The observations were analyzed using several inversion algorithms, including Milne-Eddington, LTE with 1 and 2 components, and MISMA codes. We find that the line-ratio technique applied to the 5250 Å lines is not sufficiently reliable to provide a direct magnetic diagnostic in the presence of thermal fluctuations and variable line broadening. In general, one needs to resort to inversion algorithms, ideally with realistic magnetohydrodynamic constrains. When this is done, the 5250 Å lines do not seem to provide any significant advantage over those at 6302 Å. In fact, our results point toward a better performance with the latter (in the presence of turbulent line broadening). In any case, for very weak flux concentrations, neither spectral region alone provides sufficient constraints to fully disentangle the intrinsic field strengths. Instead, we advocate for a combined analysis of both spectral ranges, which yields a better determination of the quiet-Sun magnetic properties. Finally, we propose the use of two other Fe I lines (at 4122 and 9000 Å) with identical line opacities that seem to work much better than the others.