A suspension of rodlike colloidal particles and rodlike liquid crystalline polymers is modelled as a mixture of thick (colloidal) and thin (polymeric) hard rods. Extensive immiscibility in the fluid phase is observed in the mixtures. For two species of polymers, one species much longer than the other, we observe two demixing critical points. For three polymer species we find three critical points. Polymer molecules of length I induce an attractive interaction of range I between colloidal rods. Two different polymers induce effective attractions of two different ranges. The range of the attraction determines the density at which the demixing occurs. The attractions of different ranges create demixing at different densities.