We present an algorithm for finding the smallest side length for 3 axis-aligned cubes that cover a given n-point set in d dimensions in O(6ddn log (dn)) time. This shows that the problem is fixed-parameter tractable when parameterized with d. On the other hand, using tools from parameterized complexity theory, we show that this is unlikely to be the case for 4 cubes, and with the k-center problem in the Euclidean metric, for any k>1. In particular, we prove that deciding whether a given d-dimensional set of n points can be covered by the union of 2 balls of given radius, or by 4 axis-aligned cubes of given side length, is W[1]-hard with respect to d, and thus not fixed-parameter tractable unless FPT=W[1].