Patient description: The patient is a 37-year-old woman who came to the clinic for a routine Pap smear. The test revealed stage 1B of cervical cancer. The patient tested positive for HPV 16. She has three sexual partners and four children. She underwent tubal ligation. She smokes 30 packs/year and drinks alcohol frequently. She is otherwise healthy. She was offered a radical hysterectomy.

Clinical trial eligibility criteria: eligible ages (years): 18.0-999.0,  Malignant Solid Neoplasm Metastatic Melanoma Stage III Cutaneous Melanoma AJCC v7 Stage IIIA Cutaneous Melanoma AJCC v7 Stage IIIB Cutaneous Melanoma AJCC v7 Stage IIIC Cutaneous Melanoma AJCC v7 Stage IV Cutaneous Melanoma AJCC v6 and v7 Unresectable Melanoma PHASE I 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 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 computed tomography (CT) scan, magnetic resonance imaging (MRI), or calipers by clinical exam Prior therapy is allowed; for patients enrolled in the Phase II portion of the study, patients may have received prior immunotherapy (including high-dose IL-2, ipilimumab, nivolumab, and other anti-PD1/PDL1 antibodies) or chemotherapy; however prior navitoclax, BRAF inhibitor and/or MEK inhibitor therapy will not be allowed Eastern Cooperative Oncology Group (ECOG) performance status =< 1 (Karnofsky >= 70%) Life expectancy of greater than 3 months Leukocytes >= 3,000/mcL Absolute neutrophil count >= 1 x 10^9/L Hemoglobin >= 9 g/dl (patients may be transfused to this level)  PHASE I ONLY: Patients must not have received prior navitoclax, 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 who have had immunotherapy, chemotherapy or radiotherapy within 14 days prior to the first dose of navitoclax, or prior systemic anti-cancer therapy (chemotherapy with delayed toxicity, extensive radiation therapy, immunotherapy, biologic therapy, or vaccine therapy) within the last 3 weeks prior to first dose of dabrafenib and/or trametinib; chemotherapy regimens without delayed toxicity within the last 2 weeks preceding the first dose of study treatment; biologics will not be allowed within 30 days prior to, or during, navitoclax administration Prior navitoclax, BRAF inhibitor, and MEK inhibitor is prohibited; (exceptions for Phase I are described above) Patients who are receiving any other investigational agents have received any other investigational drugs within 28 days (or five half-lives, whichever is shorter; with a minimum of 14 days from the last dose) preceding the first dose of study treatment and during the study Patients must have no clinical evidence of leptomeningeal or brain metastasis causing spinal cord compression that are symptomatic or untreated or not stable for >= 3 months (must be documented by imaging) or requiring corticosteroids; subjects on a stable dose of corticosteroids > 1 month or who have been off of corticosteroids for at least 2 weeks can be enrolled with approval of the Cancer Therapy Evaluation Program (CTEP) medical monitor; subjects must also be off of enzyme-inducing anticonvulsants for > 4 weeks History of allergic reactions attributed to compounds of similar chemical or biologic composition to navitoclax, dabrafenib, or trametinib, or excipients or to dimethyl sulfoxide (DMSO) Uncontrolled intercurrent illness including, but not limited to, ongoing or active serious infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, uncontrolled diabetes, or psychiatric illness/social situations that would limit compliance with study requirements Pregnant women are excluded from this study because navitoclax, dabrafenib, and trametinib may have teratogenic or abortifacient effects; because there is an unknown but potential risk for adverse events in nursing infants secondary to treatment of the mother with the study drugs, breastfeeding should be discontinued if the mother is treated with the study drugs Human immunodeficiency virus (HIV)-positive patients on combination antiretroviral therapy that predict to interact with any of the study drugs are ineligible because of the potential for pharmacokinetic interactions with the study drugs; appropriate studies will be undertaken in patients receiving combination antiretroviral therapy when indicated; it is not necessary to conduct HIV testing at screening; patients who are HIV-positive with undetectable viral loads, not on interacting antiretroviral therapy, and have CD4 counts above 300/mm^3 may be eligible after discussion with the principal investigator History of another malignancy; exception: patients who have been disease-free for 3 years (depending upon tumor type studied or clinical setting, 3 or 5 years can be used; e.g., for advanced melanoma and pancreatic studies 3 years is more appropriate due to aggressiveness of the disease, while 5 years can be more appropriate for prostate or ovarian cancer or adjuvant setting when life expectancy is longer), or patients with a history of completely resected non-melanoma skin cancer and/or patients with indolent secondary malignancies, are eligible; consult the CTEP medical monitor if unsure whether second malignancies meet the requirements specified above; exception: patients with history of RAS mutation-positive tumors are not eligible regardless of interval from the current study; prospective RAS testing is not required; however, if the results of previous RAS testing are known, they must be used in assessing