Bortezomib, Paclitaxel, and Carboplatin in Treating Patients With Metastatic Melanoma

Criteria: * No uncontrolled intercurrent illness including any of the following: ongoing or active infection; symptomatic congestive heart failure; unstable angina pectoris; cardiac arrhythmia * No psychiatric illness that would limit compliance with study requirements * No other uncontrolled serious medical conditions (e.g., diabetes) * No more than 1 prior cytotoxic chemotherapy regimen * No more than 2 prior immunotherapy regimens either in adjuvant or metastatic setting * At least 4 weeks since prior major radiotherapy or chemotherapy * At least 8 weeks since prior monoclonal antibody therapy * At least 4 weeks since prior immunotherapy or biologic therapy * At least 3 weeks since prior surgery * Recovered from prior therapies * No prior therapy with bortezomib, paclitaxel, or carboplatin * No other prior or concurrent chemotherapy, immunotherapy, radiotherapy, or any other therapy or supportive care considered investigational * No concurrent combination antiretroviral therapy for HIV-positive patients * No concurrent prophylactic colony-stimulating factors * Histologically confirmed malignant melanoma * Patients with significant fluid retention, including ascites or pleural effusion, may be allowed at the discretion of the principal investigator * No known brain metastases by brain imaging with contrast * Absolute neutrophil count \>= 1,500/mm\^3 * Platelet count \>= 100,000/mm\^3 * Routine urine analysis with predicted 24-hour urine protein \< 500 mg OR 1+ proteinuria by urine dipstick with 24-hour urine protein \< 500 mg * Total bilirubin \< 1.5 mg/dL * AST =\< 3 times ULN * Creatinine =\< 1.5 times ULN * ECOG performance status (PS) 0, 1, or 2 (Karnofsky PS \>= 60%) * Life expectancy by physician estimate \> 12 weeks * Not pregnant or nursing * Fertile patients must use effective contraception during and for 6 months after completion of study treatment * Negative pregnancy test * No history of allergic reactions attributed to compounds of similar chemical or biologic composition to bortezomib * No peripheral neuropathy \>= grade 2 * Manifestations of stage IV disease (e.g., cutaneous, uveal) * All melanomas, regardless of origin, allowed * Measurable disease, defined as at least one lesion whose longest diameter can be accurately measured as \>= 2.0 cm with conventional techniques or as \>= 1.0 cm with spiral CT scan * No nonmeasurable disease only, including any of the following: bone lesions, leptomeningeal disease, ascites, pleural/pericardial effusion, inflammatory breast disease, lymphangitis cutis/pulmonis, abdominal masses that are not confirmed and followed by imaging techniques, cystic lesions * Hemoglobin \>= 9.0 g/dL