  We provide bounds of Berry-Esseen type for fundamental limit theorems in
operator-valued free probability theory such as the operator-valued free
Central Limit Theorem and the asymptotic behaviour of distributions of
operator-valued matrices. Our estimates are on the level of operator-valued
Cauchy transforms and the L\'evy distance. We address the single-variable as
well as the multivariate setting for which we consider linear matrix pencils
and noncommutative polynomials as test functions. The estimates are in terms of
operator-valued moments and yield the first quantitative bounds on the L\'evy
distance for the operator-valued free Central Limit Theorem. Our results also
yield quantitative estimates on joint noncommutative distributions of
operator-valued matrices having a general covariance profile. In the
scalar-valued multivariate case, these estimates could be passed to explicit
bounds on the order of convergence under the Kolmogorov distance.
