Brief Summary: This phase I trial studies the side effects of ipilimumab and nivolumab in combination with radiation therapy, and to see how well they work in treating patients with stage II-III non-small cell lung cancer. Immunotherapy with monoclonal antibodies, such as ipilimumab and nivolumab, may help the body's immune system attack the cancer, and may interfere with the ability of tumor cells to grow and spread. Radiation therapy uses high energy rays to kill tumor cells and shrink tumors. Ipilimumab and nivolumab may also help radiation therapy work better by making tumor cells more sensitive to the radiation therapy. Giving ipilimumab and nivolumab in combination with radiation therapy may work better in treating patients with stage II-III non-small cell lung cancer compared to standard chemotherapy in combination with radiation therapy.

Detailed Description: PRIMARY OBJECTIVES: I. To determine safety and feasibility of ipilimumab-nivolumab (Bristol-Meyers-Squibb, Opdivo), a cytotoxic T-lymphocyte-associated protein 4 (CTLA4) and programmed death 1 (PD-1) inhibitor in the treatment of local-regionally advanced non-small cell lung cancer (NSCLC). SECONDARY OBJECTIVES: I. To evaluate the clinical outcomes of treatment with ipilimumab-nivolumab concurrent with radiation therapy in patients with local-regionally advanced NSCLC. II. To observe and record anti-tumor activity. III. To identify potential predictive and prognostic biomarkers for early recurrence using circulating cell-free deoxyribonucleic acid (DNA) (cfDNA). IV. To identify potential predictive and prognostic biomarkers for early recurrence using tumor tissue PD-L1 immunohistochemistry (IHC). V. To identify potential predictive and prognostic biomarkers for early recurrence using tumor tissue tumor mutational burden (TMB). VI. To identify potential resistance mechanisms using immune biomarker and genetic analysis in post-progression biopsies. EXPLORATORY OBJECTIVES: I. Tumor tissue/blood biomarkers will be assessed for tumor mutation burden (TMB) and PD-L1 immunohistochemistry. II. Patient microbiomes will be evaluated with stool specimen collection kits. OUTLINE: CONCURRENT THERAPY: Patients receive nivolumab intravenously (IV) over 30 minutes on day 1 and ipilimumab IV over 30 minutes on day 1. Treatment with nivolumab repeats every 21 days for up to 8 cycles, and treatment with ipilimumab repeats every 42 days for up to 4 cycles in the absence of disease progression or unacceptable toxicity. Within 1 day of starting nivolumab and ipilimumab, patients also undergo radiation therapy 5 days a week (Monday-Friday) over 6-7 weeks in the absence of disease progression or unacceptable toxicity. MAINTENANCE THERAPY: Patients then receive nivolumab IV over 30 minutes on day 1. Treatment repeats every 28 days for up to 8 cycles in the absence of disease progression or unacceptable toxicity. After completion of study treatment, patients are followed up every 3 months for 2 years.