Phage therapy has been viewed as a potential treatment for bacterial infections for over a century. Yet, the year 2016 marks the first phase I/II human trial of a phage therapeutic - to treat burn wound patients in Europe. The slow progress in realizing clinical therapeutics is matched by a similar dearth in principled understanding of phage therapy. Theoretical models and in vitro experiments find that combining phage and bacteria often leads to coexistence of both phage and bacteria or phage elimination altogether. Both outcomes stand in contrast to the stated goals of phage therapy. A potential resolution to the gap between models, experiments, and therapeutic use of phage is the hypothesis that the combined effect of phage and an immune system can synergistically eliminate bacterial pathogens. Here, we propose a phage therapy model that considers the nonlinear dynamics arising from interactions between bacteria, phage and the immune system. The model builds upon earlier efforts by incorporating a maximum capacity of the immune response and immune evasion by bacteria at high density. We analytically identify a synergistic regime in which phage and the immune response jointly contribute to the elimination of the target bacteria. Crucially, we find that in this synergistic regime, neither phage alone nor the immune system alone can eliminate the bacteria. We confirm these findings using numerical simulations in biologically plausible scenarios. We utilize our numerical simulations to explore the synergistic effect and its significance for guiding the use of phage therapy in clinically relevant applications.