We perform simulations of random seas based on narrow-banded spectra with directional spreading. Our wavefields are spatially homogeneous and nonstationary in time. We truncate the spectral tail for the initial conditions at different cutoff wavenumbers to assess the impact of the spectral tail on the kurtosis and spectral evolution. We consider two cases based on truncation of the wavenumber tail at and . Our simulations indicate that the peak kurtosis value increases if the tail is truncated at rather than . For the case with a wavenumber cutoff at , augmented kurtosis is accompanied by comparatively more aggressive spectral changes including redevelopment of the spectral tail. Similar trends are observed for the case with a wavenumber cutoff at , but the spectral changes are less substantial. Thus, the spectral tail appears to play an important role in a form of spectral equilibrium that reduces spectral changes and decreases the peak kurtosis value. Our findings suggest that care should be taken when truncating the spectral tail for the purpose of simulations/experiments. We also find that the equation of Fedele (2015, “On the Kurtosis of Deep-Water Gravity Waves,” J. Fluid Mech., 782, pp. 25–36) provides an excellent estimate of the peak kurtosis value. However, the bandwidth parameter must account for the spectral tail to provide accurate estimates of the peak kurtosis.