Among factors affecting the accuracy of flow simulations with Reynolds-Averaged Navier-Stokes turbulence models is modeling turbulent diffusion processes. With the use of the Gram-Charlier series expansions, the turbulent diffusion in fourth-order one-point statistical closures of the Reynolds-Averaged Navier-Stokes equations can be modeled without introducing unknown model coefficients and without assuming turbulence being Gaussian. Terms representing turbulent diffusion processes in transport equations for second- and third-order velocity correlations do not require any modeling in such closures. In this regard, fourth-order closures are a more accurate alternative to lower-order closures where turbulent diffusion is modeled on semi-empirical or Gaussian turbulence assumptions. In the current paper, the accuracy of the closing procedure based on the Gram-Charlier series expansions is evaluated using data of direct numerical simulations in an incompressible zero-pressure-gradient turbulent boundary layer over a flat plate. One-point third-, fourth-, and fifth-order velocity moments were extracted for this purpose from the dataset collected by the Fluid Dynamics Group at the Universidad Politécnica de Madrid at two streamwise locations 4101 and 5200 that correspond to channels and pipes at δ 1331 and 1626. Results demonstrate that the truncated Gram-Charlier series expansions are an accurate approximation of the fifth-order velocity moments in the considered flow.