Increasingly stringent regulations are imposed on nitrogen oxides emissions due to their numerous negative impacts on human health and the environment. Accurate, experimentally validated thermochemical models are required for the development of the next generation of combustors. This paper presents a series of experiments performed in lean, premixed, laminar, jet-wall stagnation flames at pressures of 2, 4, 8, and 16 atm. To target post-flame temperatures relevant to gas turbine engines, the stoichiometry of the non-preheated methane-air mixture is adjusted to an equivalence ratio of 0.7. One-dimensional profiles of temperature and NO mole fraction are measured via laser-induced fluorescence (LIF) thermometry and NO-LIF, respectively, to complement previously published flame speed data . The results reveal that, as the pressure increases, the maximum post-flame temperature stays relatively stable, and the concentration of NO produced through the flame front remains constant within uncertainty. Seven thermochemical models, selected for their widespread usage or recent date of publication, are validated against the experimental data. While all mechanisms accurately predict the post-flame temperature, thanks to consistent thermodynamic parameters, important disagreements are observed in the NO concentration profiles, which highlights the need to carefully select the models used as design tools. The lack of pressure dependence of NO formation that many models fail to capture is numerically investigated via sensitivity and reaction path analyses applied to the solution of flame simulations. The termolecular reaction H+O2 (+M) ↔ HO2 (+M) is shown to hinder the production of atomic oxygen and hydrogen radicals at higher pressures, which inhibits the formation of nitric oxide through the N2O pathway.