One of the delicate problems in nonparametric and high-dimensional statistics is the construction of valid inferential procedures, particularly the question of existence of `honest' confidence sets. The existing frequentist theory has highlighted several fundamental difficulties as well as some interesting ways around this problem, such as imposing shape constraints or self-similarity on the statistical parameters, or relaxing the notion of coverage probability. In contrast the recently emerged field of Bayesian non-parametrics has at hand a posterior distribution which can readily be used for inference and for the construction of credible sets, and has hence been quite popular in scientific applications and parts of machine learning. The classical justification of Bayesian procedures in parametric models is the celebrated Bernstein-von Mises theorem, which implies that posterior based inference asymptotically coincides with efficient frequentist procedures. In the nonparametric setting Freedman (Ann.Stat., 1999) has pointed out in his influential Wald lecture that the Bernstein von Mises theorem for nonparametric models does not hold true in general. We shall revisit Freedman's result and see that his findings are tied to the L^2-setting and that Bernstein-von Mises theorems can hold true in suitable infinite-dimensional spaces that have topologies that are somewhat weaker than L^2. When used with care this can give a full frequentist justification of Bayesian inferential procedures and credible sets in nonparametric models.