Nivolumab in Combination With Chemotherapy, or Nivolumab in Combination With Ipilimumab, in Advanced EGFR-Mutant or ALK-Rearranged NSCLC

Inclusion Criteria: * Histologically or cytologically confirmed advanced (stage IIIB or IV) non-small-cell lung cancer (NSCLC). * ALK-rearranged NSCLC: ALK rearrangement as assessed by ALK FISH, IHC, or next-generation sequencing (NGS). For ALK FISH, rearrangements must be detected in \>15% of tumor cells. * EGFR-mutant NSCLC: EGFR activating gene mutation (e.g., L858R, exon 19 deletion) as well as a T790M mutation per local testing. * Presence of at least one measurable lesion as defined by RECIST v1.1. A previously irradiated site lesion may only be counted as a target lesion if there is clear sign of progression since the completion of irradiation. * Prior treatment with appropriate tyrosine kinase inhibitors (TKIs) as follows: * ALK-positive NSCLC (cohorts B and D): Participants must have progressed on or after 1 or more next-generation ALK-TKI(s). * EGFR-mutant NSCLC (cohorts A and C): Participants must have progressed on or after 1 or more third-generation, T790M mutant-selective EGFR-TKI(s). * Prior systemic chemotherapy requirements are as follows: * Nivolumab plus carboplatin and pemetrexed arms (cohorts A and B): NO prior systemic chemotherapy is allowed. NOTE: Prior adjuvant or neoadjuvant chemotherapy is allowed if received more than 12 months prior to the study. * Nivolumab plus ipilimumab arms (cohorts C and D): Participants must have received a platinum-based combination chemotherapy for their advanced lung cancer and either progressed on/after this chemotherapy or are intolerant. Only ONE line of chemotherapy is allowed. NOTE: Prior adjuvant or neoadjuvant chemotherapy does not count as an additional line of chemotherapy if received more than 12 months prior to the study. * Tumor tissue sample is required following the participant's last line of systemic therapy (TKI or chemotherapy). Tissue sample may be fresh (core needle, excisional, or incisional biopsy), or archival if obtained within 6 months prior to enrollment. There can have been no systemic therapy administered after the sample was obtained. If a tissue sample is available but it has been \> 6 months and there has been no intervening therapy, the Principal Investigator may approve the sample after discussion. PD-L1 IHC testing will be performed on the tumor tissue, but positivity on the PD-L1 IHC testing is not required to enroll in the study. * Clinically asymptomatic and stable central nervous system (CNS) metastases are allowed (including untreated CNS metastases) if they have not required increasing doses of steroids or stable doses equivalent to prednisone \> 10 mg daily within 2 weeks prior to study entry for CNS symptoms. * Prior palliative radiotherapy must have been completed at least 2 weeks prior to study entry. * Subjects must have been off any prior systemic anti-cancer treatment (including TKIs) for at least 5 half-lives of that drug. * Age ≥ 18 years old. * ECOG performance status of 0 or 1. * Life expectancy ≥ 12 weeks. * Screening laboratory values must meet the following criteria: * WBC ≥ 2.0 x 109/L * Neutrophils ≥ 1.5 x 109/L * Platelet ≥ 100 x 109/L * Hemoglobin ≥ 9/dL * Serum creatinine ≤ 1.5 x ULN or calculated creatinine clearance using Cockcroft-Gault formula ≥ 50 mL/min * Female CrCl = (140-age in years) x weight in kg x 0.85 72 x serum creatinine in mg/dL * Male CrCl = (140 - age in years) x weight in kg x 1.00 72 x serum creatinine in mg/dL * Total bilirubin ≤ 1.5 x ULN (except in patients with Gilbert's syndrome who may have total bilirubin \< 3.0 mg/dL) * AST and ALT ≤ 3.0 x ULN (or ≤ 5.0 x ULN if liver metastases are present) * Females of child-bearing potential (defined as a sexually mature woman who has not undergone a hysterectomy or bilateral oophorectomy or has not been naturally post-menopausal for at least 24 consecutive months) must: * Have a negative serum or urine pregnancy test obtained within 24 hours prior to starting the investigational drug. * Not be breastfeeding. * Agree to use, and be able to comply with, highly effective contraception (failure rate less than 1% per year) without interruption while on study treatment plus 5 half-lives of nivolumab (half-life up to 25 days) plus 30 days (duration of ovulatory cycle) for a total of 23 weeks post treatment completion. Complete abstinence is acceptable if it is in line with the preferred and usual lifestyle of the patient. Period abstinence (e.g., calendar, ovulation, symptothermal, or postovulation methods) and withdrawal are not acceptable methods of contraception. Examples of non-hormonal contraceptive methods with a failure rate of \< 1% per year include bilateral tubal ligation, male sterilization, hormonal implants, established and proper use of combined oral or injected hormonal contraceptives, and certain intrauterine devices. Alternatively, two methods (e.g., two barrier methods from the options of diaphragm, cervical cap, vaginal sponge, and condom, or progesterone-only oral hormonal contraception where inhibition of ovulation is not the primary mode of action) may be combined to achieve a failure rate of \< 1% per year. Barrier methods must always be supplemented with the use of a spermicide. * Male subjects agree to remain abstinent or use a condom plus an additional contraceptive method that result in a failure rate of \<1% per year during sexual contact with a female of childbearing potential while participating in the study, during dose interruptions, and for 31 weeks following the last dose of the study treatment, even if he has undergone a successful vasectomy. Abstinence is only acceptable if it is in line with the preferred and usual lifestyle of the patient. Periodic abstinence and withdrawal are not acceptable. * Subject has the ability to understand and provide signed informed consent. * Subject has the willingness and ability to comply with scheduled visits, treatment plans, laboratory tests, and other study procedures Exclusion Criteria: * Subjects previously treated with T cell immune-modulating antibodies, including anti-CTLA-4, anti-PD-1 and/or anti-PD-L1 agents. * Subjects with symptomatic brain metastases, carcinomatous meningitis, spinal cord compression, or intractable back pain due to compression of destructive mass. * Subjects with active, known, or suspected autoimmune disease. Subjects with type I diabetes mellitus, hypothyroidism only requiring hormone replacement, skin disorders (such as vitiligo, psoriasis, or alopecia) not requiring systemic treatment, or conditions not expected to recur in the absence of an external trigger are permitted to enroll. * Subjects with a condition requiring systemic treatment with either corticosteroids (\> 10 mg daily prednisone equivalent) or other immunosuppressive medications within 14 days of randomization (unless used to treat investigational drug-related adverse events). Inhaled or topical steroids, and adrenal replacement steroid doses \> 10 mg daily prednisone equivalent, are permitted in the absence of active autoimmune disease. * Subjects with interstitial lung disease or interstitial pneumonitis that is symptomatic or may interfere with the detection or management of suspected drug-related pulmonary toxicity. * Subjects must have recovered from the effects of major surgery or significant traumatic injury at least 14 days prior to screening. No major surgery, other than diagnostic surgery, is allowed within 4 weeks prior to treatment in the study. * Co-administration of anti-cancer therapies other than those administered in the study. * Subjects with other active malignancy requiring concurrent intervention. * Known history of testing positive for human immunodeficiency virus (HIV) or known acquired immunodeficiency syndrome (AIDS). HIV-positive participants are ineligible because these participants are at increased risk of lethal infections when treated with marrow-suppressive therapy, and because there is the potential for pharmacokinetic interactions with combination antiretroviral therapy and study drugs. * Any positive test for hepatitis B virus or hepatitis C virus indicating acute or chronic infection. * Subjects with ≥ grade 2 peripheral neuropathy at enrollment per the National Cancer Institute Common Terminology Criteria for Adverse Events (NCI CTCAE). * Pregnant or lactating women. Pregnant women are excluded from this study because the study drugs (i.e., nivolumab, ipilimumab, carboplatin, and pemetrexed) have the potential for 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 these agents, breastfeeding should be discontinued. * Subjects currently enrolled in any other clinical protocol or investigational trial that involves administration of experimental therapy and/or therapeutic devices, or investigational drug. * History of allergy or hypersensitivity to any study drugs or their excipients. * Any known clinically significant concomitant medical condition, laboratory abnormality, or psychiatric illness that, in the investigator's opinion, would prevent the subject from participating in the study, pose an unacceptable risk to the patient in this study, or interfere with the interpretation of safety results.