Given a sequence ξ_1, ξ_2,... of X-valued, exchangeable random elements, let q(ξ^(n)) and p_m(ξ^(n)) stand for posterior and predictive distribution, respectively, given ξ^(n) = (ξ_1,..., ξ_n). We provide an upper bound for limsup b_n d_[[X]](q(ξ^(n)), δ_\empiricn) and limsup b_n d_[X^m](p_m(ξ^(n)), \empiricn^m), where \empiricn is the empirical measure, b_n is a suitable sequence of positive numbers increasing to +\infty, d_[[X]] and d_[X^m] denote distinguished weak probability distances on [[X]] and [X^m], respectively, with the proviso that [S] denotes the space of all probability measures on S. A characteristic feature of our work is that the aforesaid bounds are established under the law of the ξ_n's, unlike the more common literature on Bayesian consistency, where they are studied with respect to product measures (p_0)^\infty, as p_0 varies among the admissible determinations of a random probability measure.