Context. The structure of the solar chromosphere is believed to be governed by magnetic fields, even in quiet-Sun regions that have a relatively weak photospheric field. During the past decade inversion methods have emerged as powerful tools for analyzing the chromosphere of active regions. The applicability of inversions to infer the stratification of the physical conditions in a dynamic 3D solar chromosphere has not yet been studied in detail. Aims: This study aims to establish the diagnostic capabilities of non-local thermodynamical equilibrium (NLTE) inversion techniques of Stokes profiles induced by the Zeeman effect in the Ca ii λ8542 Å line. Methods: We computed the Ca ii atomic level populations in a snapshot from a 3D radiation-MHD simulation of the quiet solar atmosphere in non-LTE using the 3D radiative transfer code Multi3d. These populations were used to compute synthetic full-Stokes profiles in the Ca ii λ8542 Å line using 1.5D radiative transfer and the inversion code Nicole. The profiles were then spectrally degraded to account for finite filter width, and Gaussian noise was added to account for finite photon flux. These profiles were inverted using Nicole and the results were compared with the original model atmosphere. Results: Our NLTE inversions applied to quiet-Sun synthetic observations provide reasonably good estimates of the chromospheric magnetic field, line-of-sight velocities and somewhat less accurate, but still very useful, estimates of the temperature. Three-dimensional scattering of photons cause cool pockets in the chromosphere to be invisible in the line profile and consequently they are also not recovered by the inversions. To successfully detect Stokes linear polarization in this quiet snapshot, a noise level below 10-3.5 is necessary.