In the literature, a major discrepancy is reported between the value of the kurtosis for the normal velocity fluctuations close to the wall as found from direct numerical simulations and obtained from experiments. The origin of these high kurtosis levels is analyzed with help of a direct numerical simulation of a turbulent channel flow. In addition, a detailed analysis of LDV measurements in the near-wall region of a turbulent pipe flow is made with the results of the DNS as a starting point. In both data sets, i.e. DNS and experiments, similar velocity events were found that contribute to the high kurtosis level. The dynamics of these events can be associated with the regeneration process of streamwise vortices as described by Brooke and Hanratty [Phys. Fluids A 5, 1011 (1993)]. Based on this evidence, we conclude that the high kurtosis is of a truly physical nature. It is caused by very strong events that appear only in the near-wall region and that are rare in time as well as in space. The very rare appearance of these events explains why they are usually missed in experimental data which are mostly obtained from averages over a limited time series. In this respect the DNS results may be considered as more accurate because they are based on surface averages.