Testing the Addition of Navitoclax to the Combination of Dabrafenib and Trametinib in People Who Have BRAF Mutant Melanoma

Inclusion Criteria: * PHASE I SUBJECTS ONLY: Prior therapy is allowed; for patients enrolled in the Phase I portion of the study, patients may have received any number of prior lines of therapy including treatment with a BRAF and/or MEK inhibitor; prior navitoclax use will not be allowed, unless the patient received \< 7 days of navitoclax lead-in on this or another study and had to stop for reasons other than toxicity or disease progression * Patients must have histologically confirmed, BRAF-mutant (V600E/K) melanoma (molecularly confirmed using validated, commercially available assay performed in a Clinical Laboratory Improvement Act \[CLIA\]-approved laboratory) that is metastatic or unresectable and for which standard curative measures do not exist or are no longer effective * If test at CLIA-certified lab used a non-Food and Drug Administration (FDA) approved method, information about the assay must be provided; (FDA approved tests for BRAF V600 mutations in melanoma include: THxID BRAF Detection Kit and Cobas 4800 BRAF V600 Mutation Test) * Patients must have measurable disease, defined as at least one lesion that can be accurately measured in at least one dimension (longest diameter to be recorded for non-nodal lesions and short axis for nodal lesions) as \>= 20 mm with conventional techniques or as \>= 10 mm with spiral CT scan, MRI, or calipers by clinical exam * Prior therapy is allowed; for patients enrolled in the Phase II portion of the study, patients may ha…