Dabrafenib and Trametinib Followed by Ipilimumab and Nivolumab or Ipilimumab and Nivolumab Followed by Dabrafenib and Trametinib in Treating Patients With Stage III-IV BRAFV600 Melanoma

Inclusion Criteria: * STEP 1 * Age \>= 18 years. Because no dosing or adverse event data are currently available on the use of dabrafenib or dabrafenib + trametinib or nivolumab or nivolumab + ipilimumab therapy in patients \< 18 years of age, children are excluded from this study, but will be eligible for future pediatric trials * Eastern Cooperative Oncology Group (ECOG) performance status: 0 or 1 * Women must not be pregnant or breast-feeding, as the effects of ipilimumab + nivolumab or dabrafenib + trametinib on the developing human fetus are unknown * All females of childbearing potential must have a blood test or urine study within 2 weeks prior to registration to rule out pregnancy * A female of childbearing potential is anyone, regardless of whether they have undergone tubal ligation, who meets the following criteria: 1) has not undergone a hysterectomy or bilateral oophorectomy; or 2) has not been naturally postmenopausal for at least 24 consecutive months (i.e., has had menses at any time in the preceding 24 consecutive months) * The effects of dabrafenib and trametinib or ipilimumab and nivolumab on the developing human fetus are unknown; furthermore, dabrafenib has been reported to interfere with the effect of hormone based oral contraceptives; for this reason and because other therapeutic agents used in this trial are known to be teratogenic, women of child-bearing potential and sexually active males must agree to use at least two other accepted and effecti…