We exhibit a class of Schottky subgroups of $\mathbf{PU}(1,n)$ ($n \geq 2$) which we call well-positioned and show that the Hausdorff dimension of the limit set $��_��$ associated with such a subgroup $��$, with respect to the spherical metric on the boundary of complex hyperbolic $n$-space, is equal to the growth exponent $��_��$. For general $��$ we establish (under rather mild hypotheses) a lower bound involving the dimension of the Patterson-Sullivan measure along boundaries of complex geodesics. Our main tool is a version of the celebrated Ledrappier-Young theorem.