Standard-Dose Combination Chemotherapy or High-Dose Combination Chemotherapy and Stem Cell Transplant in Treating Patients with Relapsed or Refractory Germ Cell Tumors

1. Documentation of Disease * Histologic Documentation: Confirmation of GCT histology (both seminoma and nonseminoma) on pathologic review at the center of enrollment. * Tumor may have originated in any primary site. NOTE: In rare circumstances, patients will be allowed to enroll even if a pathologic diagnosis may not have been established. * This would require a clinical situation consistent with the diagnosis of GCT (testicular, peritoneal, retroperitoneal or mediastinal mass, elevated tumor marker levels {HCG ≥ 500; AFP ≥ 500} and typical pattern of metastases) 2. Evidence of Disease * Must have evidence of progressive or recurrent GCT (measurable or non-measurable) following one line of cisplatin-based chemotherapy, defined as meeting at least one of the following criteria: * Tumor biopsy of new or growing or unresectable lesions demonstrating viable non-teratomatous GCT (enrollment on this study for adjuvant treatment after macroscopically complete resection of viable GCT is not allowed). In the event of an incomplete gross resection where viable GCT is found, patients will be considered eligible for the study. * Consecutive elevated serum tumor markers (HCG or AFP) that are increasing. Increase of an elevated LDH alone does not constitute progressive disease. * Development of new or enlarging lesions in the setting of persistently elevated HCG or AFP, even if the HCG and AFP are not continuing to increase. 3. Prior Treatment * Must have received 3-6 cycles of cisplatin-based chemotherapy as part of first-line (initial) chemotherapy. * Prior POMBACE, CBOP-BEP, or GAMEC are allowed. * Note: For patients requiring immediate treatment, 1 cycle of conventional-dose salvage chemotherapy is allowed. Therefore, these patients may have received 7 prior cycles of chemotherapy. 6 cycles as part of first-line chemotherapy and 1 cycle of salvage conventional chemotherapy. * No more than one prior line of chemotherapy for GCT (other than the 1 cycle of salvage chemotherapy as defined in the protocol) * Definition of one line of chemotherapy: One line of therapy can in some cases consist of 2 different cisplatin-based treatment combinations, provided there is no disease progression between these two regimens. * Prior treatment with carboplatin as adjuvant therapy is allowed, provided patients meet other eligibility criteria (e.g., the patient has also received 3-4 cycles of cisplatin-based chemotherapy). * Prior treatment with 1-2 cycles of BEP or EP as adjuvant chemotherapy for early stage GCT is allowed, provided the patient also received 3-4 cycles of BEP or EP again at relapse. Patients treated with 3-4 cycles of VIP at relapse following 1-2 cycles of BEP/EP are not eligible as this would be considered more than 1 line of prior therapy. * No prior treatment with high-dose chemotherapy (defined as treatment utilizing stem cell rescue) * No prior treatment with TIP with the exception when given as a bridge to treatment on protocol for patients with rapidly progressive disease who cannot wait to complete the eligibility screening process. Only one cycle is allowed. * No concurrent treatment with other cytotoxic drugs or targeted therapies. * No radiation therapy (other than to the brain) within 14 days of day 1 of protocol chemotherapy except radiation to brain metastases, which must be completed 7 days prior to start of chemotherapy. * No previous chemotherapy within 17 days prior to enrollment. A minimum of three weeks after the last day of the start of the previous chemotherapy regimen before the first day of chemotherapy on study protocol. * Must have adequate recovery from prior surgery (eg, healed scar, resumption of diet) 4. Age ≥ 14 years (≥ 18 years in Germany) 5. ECOG Performance Status 0 to 2 6. Male gender 7. Required Initial Laboratory Values: * Absolute Neutrophil Count (ANC) ≥ 1,500/mm\^3 * Platelet Count ≥ 100,000/mm\^3 * Calculated creatinine clearance ≥ 50 mL/min * Bilirubin ≤ 2.0 x upper limits of normal (ULN) * AST/ALT ≤ 2.5 x upper limits of normal (ULN) 8. No concurrent malignancy other than non-melanoma skin cancer, superficial noninvasive (pTa or pTis) TCC of the bladder, contralateral GCT, or intratubular germ cell neoplasia. Patients with a prior malignancy, but at least 2 years since any evidence of disease are allowed. 9. Negative Serology (antibody test) for the following infectious diseases: * Human Immunodeficiency Virus (HIV) type 1 and 2 * Human T-cell Leukemia Virus (HTLV) type 1 and 2 (mandatory in US but optional in Canada and Europe) * Hepatitis B surface antigen * Hepatitis C antibody 10. No late relapse with completely surgically resectable disease. Patients with late relapses (defined as relapse ≥ 2 years from the date of completion of the last chemotherapy regimen) whose disease is completely surgically resectable are not eligible. Patients with late relapses who have unresectable disease are eligible. 11. No large (≥ 2 cm) hemorrhagic or symptomatic brain metastases until local treatment has been administered (radiation therapy or surgery). Treatment may begin ≥ 7 days after completion of local treatment. Patients with small (\< 2 cm) and asymptomatic brain metastases are allowed and may be treated with radiation therapy and/or surgery concurrently with Arm A or cycles 1 and 2 of Arm B if deemed medically indicated. Radiation therapy should not be given concurrently with high-dose carboplatin or etoposide. 12. No secondary somatic malignancy arising from teratoma (e.g., teratoma with malignant transformation) when it is actively part of the disease recurrence or progression.