We make a detailed investigation of all spaces Q_{n₁···n_N}^{q₁···q_N} of the form of U(1) bundles over arbitrary products ∏_iCP^n_i of complex projective spaces, with arbitrary winding numbers q_i over each factor in the base. Special cases, including Q₁₁¹¹ (sometimes known as T¹¹), Q₁₁₁¹¹¹ and Q₂₁³², are relevant for compactifications of type IIB and D = 11 supergravity. Remarkable `conspiracies' allow consistent Kaluza-Klein S⁵, S⁴ and S⁷ sphere reductions of these theories that retain all the Yang-Mills fields of the isometry group in a massless truncation. We prove that such conspiracies do not occur for the reductions on the Q_{n₁···n_N}^{q₁···q_N} spaces, and that it is inconsistent to make a massless truncation in which the non-Abelian SU(n_i + 1) factors in their isometry groups are retained. In the course of proving this we derive many properties of the spaces Q_{n₁···n_N}^{q₁···q_N} of more general utility. In particular, we show that they always admit Einstein metrics, and that the spaces where q_i = (n_i + 1)/ℓ all admit two Killing spinors. We also obtain an iterative construction for real metrics on CPⁿ, and construct the Killing vectors on Q_{n₁···n_N}^{q₁···q_N} in terms of scalar eigenfunctions on CP^n_i. We derive bounds that allow us to prove that certain Killing-vector identities on spheres, necessary for consistent Kaluza-Klein reductions, are never satisfied on Q_{n₁···n_N}^{q₁···q_N}.