Yes, there is a central limit theorem that guarantees convergence in probability to the Wigner semicircle law, see for example Central limit theorem for linear eigenvalue statistics of random matrices with independent entries., or Chapter 2 of Tao's Topics in random matrix theory.
The convergence to the semicircle law when the matrix dimension $n\rightarrow\infty$ requires that the variances $\sigma_{ij}^2$ of the off-diagonal matrix elements are comparable, meaning that $$0<c_{\rm inf}\leq n\sigma_{ij}^2\leq c_{\rm sup},\;\;1\leq i<j\leq n,$$ for $n$-independent constants $c_{\rm inf}$ and $c_{\rm sup}$, see Universality of Wigner random matrices: a Survey of Recent Results.
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 Limiting eigenvalue distribution for band random matrices.