  M dwarfs are key targets for high-resolution spectroscopic analyses due to a
high incidence of these stars in the solar neighbourhood and their importance
as exoplanetary hosts. Several methodological challenges make such analyses
difficult, leading to significant discrepancies in the published results. We
compare M dwarf parameters derived by recent high-resolution near-infrared
studies with each other and with fundamental stellar parameters. We also assess
to what extent deviations from local thermodynamic equilibrium (LTE) for Fe and
K influence the outcome of these studies. We carry out line formation
calculations based on a modern model atmosphere grid along with a synthetic
spectrum synthesis code that treats formation of atomic and molecular lines in
cool-star atmospheres including departures from LTE. We use near-infrared
spectra collected with the CRIRES instrument at the ESO VLT as reference
observational data. We find that the effective temperatures obtained by the
different studies mostly agree to better than 100 K. We see a much worse
agreement in the surface gravities and metallicities. We demonstrate that
non-LTE effects are negligible for Fe I in M-dwarf atmospheres but are
important for K I. These effects, leading to K abundance and metallicity
corrections on the order of 0.2 dex, may be responsible for some of the
discrepancies in the published analyses. Differences in the
temperature-pressure structures of the atmospheric models may be another factor
contributing to the discrepancies, in particular at low metallicities and high
effective temperatures. In high-resolution spectroscopic studies of M dwarfs
attention should be given to details of the line formation physics as well as
input atomic and molecular data. Collecting high-quality, wide wavelength
coverage spectra of benchmark M dwarfs is an essential future step.
