This study is researching an experimental drug called linvoseltamab ("study drug"). This study is focused on patients who have AL amyloidosis that has returned or have failed other therapies and need to be treated again. The study consists of 2 phases (Phase 1 and Phase 2): * In Phase 1, linvoseltamab will be given to a small number of participants to study the side effects of the study drug and to determine the recommended doses of the study drug to be given to participants in Phase 2. * In Phase 2, linvoseltamab will be given to more participants to continue to assess the side effects of the study drug and to evaluate the ability of linvoseltamab to treat AL amyloidosis. The study is looking at several other research questions, including: * How many participants treated with linvoseltamab have improvement in the abnormal proteins that cause organ problems and for how long * How many participants treated with linvoseltamab have improvement in the heart or kidney and for how long * What the right dosing regimen is for linvoseltamab * What side effects may happen from taking linvoseltamab * How much linvoseltamab is in the blood at different times * Whether the body makes antibodies against linvoseltamab (which could make the drug less effective or could lead to side effects)