The rise of pervasive antibiotic resistance has led to a renewal of interest in using bacteriophage (phage) to treat bacterial infections. Phage therapy has been viewed as a potential treatment for over a century. Yet this year 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 theoretical understanding of how, why, and when phage therapy works. In contrast to the stated goals of phage therapy, standard phage-bacteria models and in vitro experiments often lead to coexistence of phage and bacteria. Instead, it has been hypothesized that phage together with an immune system can synergistically eliminate a bacterial pathogen. Existing models of bacteria-phage-immune system dynamics include simplified assumptions such as unbounded growth of the immune response or the lack of immune evasion by bacteria. Moreover, they do not provide a mechanistic basis for synergistic effectiveness of phage therapy. Here, we propose a model of phage therapy that incorporates a maximum capacity of the immune response and immune evasion by bacteria at high density. We identify a synergistic regime in which phage and the immune response jointly contribute to the elimination of the bacteria. Crucially, we find that in this regime, neither phage alone nor the immune system alone can eliminate the bacteria. We study the mechanism underlying the synergistic effect and its significance for different infection and immune parameters of potential clinical relevance.