Yes, there is a central limit theorem that guarantees convergence in probability to the Wigner semicircle law, see for example <a href="http://arxiv.org/abs/0809.4698">Central limit theorem for linear eigenvalue statistics of random matrices with independent entries.</a> One important class of random matrices where the universality breaks down is the class of banded random matrices, where the nonzero matrix elements are those near the diagonal, see for example <A HREF="http://math.iisc.ernet.in/~manju/RMT/Molchanov%20Pastur%20Khorunzi.pdf">Limiting eigenvalue distribution for band random matrices.</A>