Bolthausen established a bound of order $1/\sqrt{n}$ on the rate
of convergence in the central limit theorem for martingale difference arrays
having bounded conditional moments of order 4. In the present paper it is shown
how much this moment condition can be relaxed while maintaining the same rate
of convergence. An example shows that, unlike in the i.i.d. case, a moment
condition of order 3 is not enough. Furthermore, exact rates of convergence are
derived for moment conditions of order between 2 and 3.