Simple Summary: Chimeric antigen receptor (CAR) T-cell therapy has shown much promise in liquid tumors but often fails in solid tumors. This work uses a computational model to examine under what conditions this therapy might fail or be successful. The model includes interactions between cancer cells, CAR T-cells (treatment), and vascular cells (that feed and support tumor growth). From our results, we determined specific tumor conditions in which CAR T-cell therapy is predicted to fail and suggest a combination treatment that might improve the efficacy of the treatment. Chimeric antigen receptor (CAR) T-cell therapy has been successful in treating liquid tumors but has had limited success in solid tumors. This work examines unanswered questions regarding CAR T-cell therapy using computational modeling, such as, what percentage of the tumor must express cancer-associated antigens for treatment to be successful? The model includes cancer cell and vascular and CAR T-cell modules that interact with each other. We compare two different models of antigen expression on tumor cells, binary (in which cancer cells are either susceptible or are immune to CAR T-cell therapy) and gradated (where each cancer cell has a probability of being killed by a CAR T-cell). We vary the antigen expression levels within the tumor and determine how effective each treatment is for the two models. The simulations show that the gradated antigen model eliminates the tumor under more parameter values than the binary model. Under both models, shielding, in which the low/non-antigen-expressing cells protect high antigen-expressing cells, reduced the efficacy of CAR T-cell therapy. One prediction is that a combination of CAR T-cell therapies that targets the general population of cells as well as one that specifically targets cancer stem cells should increase its efficacy. [ABSTRACT FROM AUTHOR]