This paper describes a bundle method for (approximately) minimizing complicated nonsmooth convex functions with additive structure, with the primary goal of computing bounds on the solution values of difficult optimization problems such as stochastic integer programs. The method combines features that have appeared in previously proposed bundle methods, but not in the particular configuration we propose. In particular, we use a disaggregate cutting-plane model and approximate lower oracles with on-demand accuracy, and we design the method so that certain trial points can be abandoned without having to invest the full effort of evaluating all their subproblems.