We study the phenomenon of jamming in driven diffusive systems. We introduce a simple microscopic model in which jamming of a conserved driven species is mediated by the presence of a non-conserved quantity, causing an effective long-range interaction of the driven species. We study the model analytically and numerically, providing strong evidence that jamming occurs; however, this proceeds via a strict phase transition (with spontaneous symmetry breaking) only in a prescribed limit. Outside this limit, the nearby transition (characterised by an essential singularity) induces sharp crossovers and transient coarsening phenomena. We discuss the relevance of the model to two physical situations: the clustering of buses, and the clogging of a suspension forced along a pipe.