We examine the changing immiscibility of Widom-Rowlinson-type model mixtures as the number of components is varied. In these mixtures, molecules interact with molecules of a different component via a hard-core interaction but do not interact with other molecules of the same component at all. The interaction between any two molecules of different components is the same, it is that between two parallel hard (hyper-)cubes. Theoretical calculations show that an equimolar mixture of less than 31 components demixes in the fluid phase but if there are 31 or more components demixing is preempted by solidification. These calculations are exact in the limit of infinite dimensionality. The switchover from demixing to solidification as the number of components increases is due to the high entropy cost incurred by a mixture of many components phase separating into a large number of pure phases. A Widom-Rowlinson mixture is shown to be (nearly) equivalent to the Zwanzig model of a liquid crystal at a dimensionality equal to the number of components in the mixture. Using the results obtained for the mixtures we speculate that there is an upper bound to the dimensionality at which liquid crystallinity can exist.