  Let $X_1,\ldots,X_N$ be i.i.d.\ random variables distributed like $X$.
Suppose that the first $k \geq 3$ moments $\{ \mathbb{E}[X^j] : j =
1,\ldots,k\}$ of $X$ agree with that of the standard Gaussian distribution,
that $\mathbb{E}[|X|^{k+1}] < \infty$, and that there is a subinterval of
$\mathbb{R}$ of width $w$ over which the law of $X$ has a density of at least
$h$. Then we show that \begin{align} \label{eq:bnew} \sup_{s \in \mathbb{R}}
\left| \mathbb{P} \left( \frac{X_1 + \ldots + X_N}{ \sqrt{N} } \leq s \right) -
\int_{-\infty}^s \frac{ e^{ - u^2/2} \mathrm{d} u }{ \sqrt{2 \pi }} \right|
\leq 3 \left\{ \frac{\mathbb{E}[|X|^{k+1}]}{ N^{ \frac{k-1}{2}} } + e^{ - c
hw^3 N/\mathbb{E}[|X|^{k+1}] } \right\}, \end{align} where $c > 0$ is
universal. By setting $k=3$, we see that in particular all symmetric random
variables with densities and finite fourth moment satisfy a Berry-Esseen
inequality with a bound of the order $1/N$.
  Thereafter, we study the Berry-Esseen theorem as it pertains to perturbations
of the Bernoulli law with a small density component, showing by means of a
reverse inequality that the power $hw^3$ in the exponential term is
asymptotically sharp.
