Modified Immune Cells (CD19-CD22 CAR T Cells) in Treating Patients With Recurrent or Refractory CD19 Positive, CD22 Positive Leukemia or Lymphoma

This phase I/II trial studies the side effects and best dose of modified immune cells called CD19-CD22 chimeric antigen receptor (CAR) T cells in treating patients with CD19 positive(+), CD22+ B-acute lymphoblastic leukemia, chronic lymphocytic leukemia, or non-Hodgkin's lymphoma that has come back (recurrent) or does not respond to treatment (refractory). T-cells are collected from the patient and genetic materials called "chimeric antigen receptors (CAR)" are transferred to the collected T-cells. The CAR T-cells are then infused back to the patient's body. Giving CD19- CD22 CAR T cells after chemotherapy may help to control the disease.