This phase I trial studies the side effects of huJCAR014 in treating patients with relapsed or refractory B-cell non-Hodgkin lymphoma or acute lymphoblastic leukemia. huJCAR014 CAR-T cells are made in the laboratory by genetically modifying a patient's T cells and may specifically kill cancer cells that have a molecule CD19 on their surfaces. In Stage 1, dose-finding studies will be conducted in 3 cohorts: 1. Aggressive B cell NHL 2. Low burden ALL 3. High burden ALL In Stage 2, studies may be conducted in one or more cohorts to collect further safety, PK, and efficacy information at the huJCAR014 dose level(s) selected in Stage 1 for the applicable cohort(s). There are two separate cohorts for stage 2: 1. Cohort 2A, CAR-naïve (n=10): patients who have never received CD19 CAR-T cell therapy. 2. Cohort 2B, CAR-exposed (n=27): patients who have previously failed CD19 CAR-T cell therapy.