This phase I/II trial studies the side effects and best dose of bosutinib when given together with inotuzumab ozogamicin and to see how well it works in treating patients with acute lymphoblastic leukemia or chronic myeloid leukemia that has come back or does not respond to treatment. Bosutinib may stop the growth of cancer cells by blocking some of the enzymes needed for cell growth. Immunotoxins, such as inotuzumab ozogamicin, are antibodies linked to a toxic substance and may help find cancer cells that express CD22 and kill them without harming normal cells. Giving bosutinib together with inotuzumab ozogamicin may be a better treatment for acute lymphoblastic leukemia or chronic myeloid leukemia.
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Maximum Tolerated Dose (MTD) of Bosutinib All Phase I Participants
Timeframe: At day 28
Number of Participants With a Major Hematologic Response
Timeframe: Up to 6 years, 11 months