In 1957, Adams gave the first example of two different homotopy types, say X and Y, whose Postnikov approximations, X(n) and Y(n), are homotopy equivalent for each n. He did this in response to a question posed by J.H.C.Whitehead. Adams gave an explicit description of both spaces and showed they are different, up to homotopy, by noting that one contains a sphere as a retract whereas the other does not, [1]. Recently, in our study of infinite dimensional spaces, we have had to confront the same problem. Often we can prove that for a given space, e.g., X = S3 × K(Z, 3), there are many other spaces, up to homotopy, with the same n-type as X for all n. But when asked to describe one of them, we had to plead ignorance. To correct this situation we began to look for explicit descriptions and for homotopy invariants that are not determined by finite approximations. In doing so, we found some new examples and a new answer, involving automorphism groups, to the question posed in the title. This paper deals with those examples. We find them intriguing, in part, because they indicate an unexpected lack of homogeneity among spaces of the same n-type for all n. Nevertheless, as examples go, Adams's original one remains one of our favorites because of its simplicity and explicit nature. We commend it to the reader.