Let $X_1,\dots$ be a sequence of independent random variables with a common distribution function $V(x)$. Put \begin{gather*} F_n(x)=\mathbf P\biggl\{\frac{1}{b_n}(X_1+\dots+X_n)-a_n<x\biggr\},\\ \Phi(x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^x e^{t^2/2}\,dt,\quad\Delta(b_n,a_n)=\sup_x|F_n(x)-\Phi(x)|,\\ \Delta_n=\inf_{a_n,b_n}\Delta(b_n,a_n) \end{gather*} where $a_n$, $b_n$ ($b_n>0$) are sequences of real numbers. The paper deals with questions of the accuracy in estimating $|F_n(x)-\Phi(x)|$ when $V(x)$ belongs to the domain of attraction of a normal law. In particular, necessary and sufficient conditions for $$ \biggl(\sum_{n=1}^{\infty}(g(n)\Delta_n)^s\frac{1}{n}\biggr)^{1/s}<\infty,\qquad 1\le s\le\infty, $$ are obtained. (Here $g(x)$ is a function which satisfies some conditions.)