Additional Study Information: This is a 2-part, multicenter study of a study drug in pediatric and young adult patients with mature B-cell non-Hodgkin Lymphoma which has come back or has not responded to the standard therapy. The study treatment is considered an experimental because the study drug is not approved by the United States (US) Food and Drug Administration (FDA) for treating children with mature B-cell non-Hodgkin Lymphoma. The purpose of this study is to see whether B-cell Non-Hodgkin Lymphoma responds to the study drug when taken with a standard chemotherapy regimen of either rituximab, ifosfamide, carboplatin, etoposide (RICE) or rituximab, vincristine, ifosfamide, carboplatin, idarubicin (RVICI), and the effects (good and bad) of the treatment. The safety of the study drug will also be studied. This study is divided into 2 parts, Part 1 and Part 2. Part 1 has closed to enrollment. In Part 2, subjects will be assigned by chance (like flipping a coin) to one of the two groups: 1) RICE or RVICI standard chemotherapy + study drug 2) RICE or RVICI standard chemotherapy only.