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The iceberg cubing problem is to compute the multidimensional group-by partitions that satisfy given aggregation constraints. Pruning unproductive computation for iceberg cubing when nonantimonotone constraints are present is a great challenge because the aggregate functions do not increase or decrease monotonically along the subset relationship between partitions. In this paper, we propose a novel bound prune cubing (BP-Cubing) approach for iceberg cubing with nonantimonotone aggregation constraints. Given a cube over n dimensions, an aggregate for any group-by partition can be computed from aggregates for the most specific n--dimensional partitions (MSPs). The largest and smallest aggregate values computed this way become the bounds for all partitions in the cube. We provide efficient methods to compute tight bounds for base aggregate functions and, more interestingly, arithmetic expressions thereof, from bounds of aggregates over the MSPs. Our methods produce tighter bounds than those obtained by previous approaches. We present iceberg cubing algorithms that combine bounding with efficient aggregation strategies. Our experiments on real-world and artificial benchmark data sets demonstrate that BP-Cubing algorithms achieve more effective pruning and are several times faster than state-of-the-art iceberg cubing algorithms and that BP-Cubing achieves the best performance with the top-down cubing approach.