Records in a database are often considered as points in a space of dimension equal to the number of data fields. Then the design of certain data structures leads to questions about partitions of the space. A set S of n points in d-space is said to be &agr;(n)-partitionable if there exists a set of d mutually intersecting hyperplanes that devide d-space into 2^d open regions such that no 2^d − 1 regions contain more than &agr;(n) points of S. Every planar set S is 3-4- n-partitionable and every set in 3-space is 23-24- n-partitionable; this was proved by Willard [1] and Yao [2], respectively. In this paper, it is shown that there exist sets S of arbitrary cardinality in d-space, d ≥5, for which d² + 1 open regions together contain at least n − d² points of S, in any partition by d intersecting hyperplanes. Moreover, at least 2^d − d² − 1 open regions contain no points of S. Thus, the effective balanced-tree approaches of the cited authors may not be generalized to dimensions d ≥5.