This trial tests the safety and best dose of a new treatment using modified immune cells (CAR T-cells) for patients with certain recurring or hard-to-treat lymphoid cancers. The treatment involves giving patients a brief period of chemotherapy followed by an infusion of these specially designed cells to target and kill cancer cells. Anti-CD19 CAR T-cells currently represent transformational therapy for relapsed/refractory aggressive B-cell lymphomas where durable remissions can be induced in patients with previously incurable chemotherapy-refractory disease.