The verification of statistical models of sea surface elevations based on the decomposition of the wave profile into degrees of a small parameter (wave steepness) and in terms of multidimensional integrals of wave spectra was carried out. For verification, wave measurement data were used to calculate the skewness and excess kurtosis of surface elevations, as well as the distribution of crests and troughs. Two factors are identified that limit the use of estimates of skewness Aη and excess kurtosis Eη obtained from existing models. First, the model estimates Aη and Eη are always non-negative, although the measurement data show that the lower limit of the ranges in which the skewness and excess kurtosis change is in the region of negative values. Secondly, almost all existing models are one-parameter models, using wave steepness and wave age as predictors; whereas the measured data indicate that there is no clear relationship. The values of Aη and Eη vary greatly for fixed values of the predictors. Existing statistical models can only describe average changes Aη and Eη. This limits the scope of their application. The analysis of the probability density functions of the troughs FT h and crests FCr showed that the function calculated for Aη < 0 in the region above the distribution mode exceeds the values corresponding to the Rayleigh distribution, and the relationship FT h ≈ FCr holds. The second order nonlinear model is inconsistent with this result. Negative skewness values are observed much less frequently than positive ones, so the functions FT h and FCr calculated for the whole ensemble of situations are consistent with the second-order nonlinear model.