In this talk, we study higher-order accuracy of Berry-Esseen type and bootstrap approximations for a distribution of a smooth function of a sample average in high-dimensional non-asymptotic framework. The considered higher-order Berry-Esseen type inequalities extend the classical normal approximation bounds. These results justify in non-asymptotic setting that the bootstrapping can outperform Gaussian (or chi-squared) approximation in accuracy with respect to the dimension and the sample size. Furthermore, we show that the obtained critical ratios between the dimension and the sample size cannot be improved under imposed conditions on moments of the distributions. In addition, the presented results lead to improvements of accuracy of a bootstrap procedure for general log-likelihood ratio statistics (under certain regularity conditions).