Random mesh phases share many common features with the classical lamellar phase in that they are layered phases; but crucially, they possess nonuniform interfacial curvature, since the lamellae are pierced by water-filled pores. The introduction of curvature into the lamellae has been posited as a transitional precursor for other lyotropic phases. In this paper, we show that simple H-2 nuclear magnetic resonance (NMR) experiments provide strong indication for the formation of the random mesh phase and the NMR data correlate well with literature results from small-angle X-ray scattering. The thermal evolution of the recorded quadrupolar splitting (Delta v(Q)) is monitored within the lamellar phase of two nonionic surfactants, C16E6 and C12E5, as the samples are cooled or heated, and a marked and reversible change in the evolution of Delta v(Q) is observed. Data from heavy water and deuterium labeled surfactant show the same temperature dependence and consequently report on the same structural changes with temperature. The formation of the random mesh phase is quantified in terms of an effective order parameter that is unity in the classical lamellar phase and takes values of < 1 in the random mesh phase, reaching 0.6 at lower temperatures.