Nivolumab and Relatlimab in Treating Participants With Advanced Chordoma

Inclusion Criteria: * Written informed consent must be obtained from the subject/legal representative prior to performing any protocol-related procedures, including screening evaluations. * Capable of providing informed consent and complying with trial procedures. * Eastern Cooperative Oncology Group performance status (ECOG PS) 0-1. * Life expectancy \> 12 weeks. * Histological confirmation of chordoma. * Adequate archival tissue must be available within 6 months prior to signing consent. If not, an adequate tumor specimen obtained by either excisional biopsy, incisional biopsy or core needle biopsy must be sent to the central pathology lab for evaluation. The material must measure at least 0.8 x 0.1 cm in size or contain at least 50 tumor cells. * Locally advanced, unresectable, and/or metastatic chordoma with evidence of disease progression by either computed tomography (CT) or magnetic resonance imaging (MRI) scan, or loss of neurologic function on or after the last cancer therapy within 6 months prior to randomization. * Measurable tumor lesions according to Response Evaluation Criteria in Solid Tumors (RECIST) 1.1 criteria. * Women must not be able to become pregnant (e.g., post-menopausal for at least 1 year, surgically sterile, or practicing adequate birth control methods) for the duration of the study. (Adequate contraception includes: oral contraception, implanted contraception, intrauterine device implanted for at least 3 months, or barrier method in conjunction with spermicide.) * Women must not be breastfeeding. * Males and their female partner(s) of child-bearing potential must use 2 forms of effective contraception (condom or vasectomy for males) from the last menstrual period of the female partner during the study treatment and agree to continue use for 6 months after the final dose of study treatment. * Reproductive status: * Women of childbearing potential (WOCBP) must have a negative serum or urine pregnancy test (minimum sensitivity 25 IU/L or equivalent units of human chorionic gonadotropin \[HCG\]) at screening. * Women must not be breastfeeding. * Women of childbearing potential (WOCBP) must agree to follow instructions for method(s) of contraception for the duration of treatment with study treatments plus 24 weeks after the last dose of study treatment. * Males who are sexually active with WOCBP must agree to follow instructions for method(s) of contraception for the duration of treatment with study treatments plus 33 weeks after the last dose of study treatment. Male participants must be willing to refrain from sperm donation during this time. * Azoospermic males are exempt from contraceptive requirements. WOCBP who are continuously not heterosexually active are also exempt from contraceptive requirements, and still must undergo pregnancy testing as described in this section. * Male subjects must be willing to refrain from sperm donation during the entire study and for 5 half-lives of study drug plus 90 days (duration of sperm turnover) XX days after dosing has been completed. * Investigators shall counsel WOCBP, and male subjects who are sexually active with WOCBP, on the importance of pregnancy prevention and the implications of an unexpected pregnancy. Investigators shall advise on the use of highly effective methods of contraception, which have a failure rate of \< 1% when used consistently and correctly. * Hemoglobin \>= 8 g/dL. * Absolute neutrophil count \>= 1,500/mm\^3. * Platelet count \>= 75,000/mm\^3. * Aspartate aminotransferase (AST) and alanine aminotransferase (ALT) =\< 2.5 ? institutional upper limit of normal (ULN); for subjects with liver metastases, AST or ALT =\< 5 ? ULN. * Bilirubin =\< 1.5 x ULN; for subjects with documented/suspected Gilbert?s disease, bilirubin =\< 3 x ULN. * Left ventricular ejection fraction (LVEF) assessment with documented LVEF \>= 50% by either transthoracic echocardiography (TTE) or multiple-gated acquisition (MUGA) (TTE preferred test) within 6 months from first study drug administration. * Willingness to provide consent for biopsy samples. Tumor biopsies will be required for all subjects. Tumor lesions used for biopsy should not be lesions used as RECIST target lesions, unless there are no other lesions suitable for biopsy. If a RECIST target lesion is used for biopsy the lesion must be \>= 2 cm in longest diameter. Exclusion Criteria: * Palliative surgery and/or radiation treatment within 28 days prior to date of randomization; also no steroids are permitted as of 28 days of starting the study. * Inability to give informed consent. * Exposure to any therapeutic agent (investigational or conventional) within 7 days of date of randomization or to any agent for which 5 half lives have not elapsed. * An inadequate tumor specimen as defined by the central pathologist. * History of other malignancies except cured basal cell carcinoma, cutaneous squamous cell carcinoma, melanoma in situ, superficial bladder cancer or carcinoma in situ of the cervix; for other malignancies, must be documented to be free of cancer for \>= 2 years. All other cases can be considered on a case by case basis at the discretion of the principal investigator. * Current, serious, clinically significant cardiac arrhythmias or hypertension that is not adequately controlled, per investigator discretion. * Concomitant use of medications associated with a high incidence of QT prolongation will require clearance by medical monitor. * Active, clinically significant serious infection requiring treatment with antibiotics, anti-virals or anti-fungals. The Medical Monitor should be contacted for any uncertainties. * Any condition that might interfere with the subject?s participation in the study, safety, or in the evaluation of the study results. * Concurrent enrollment in another clinical study, unless it is an observational (non-interventional) clinical study or the follow-up period of an interventional study. * Prior exposure to any anti-LAG3 antibodies. (prior PD-1/PD-L1 antibodies are permitted). * Any concurrent chemotherapy, immunotherapy, biologic or hormonal therapy for cancer treatment. Concurrent use of hormones for non-cancer-related conditions (eg, insulin for diabetes and hormone replacement therapy) is acceptable. * Current or prior use of immunosuppressive medication within 28 days before the first dose of nivolumab, with the exceptions of intranasal, topical, and inhaled corticosteroids or systemic corticosteroids at physiologic doses not to exceed 10 mg/day of prednisone or equivalent (use for brain metastases is not permitted 28 days prior to start of therapy). * Active or prior documented autoimmune disease within the past 3 years. * Note: Subjects with active, known or suspected autoimmune disease such as vitiligo, type I diabetes mellitus, residual hypothyroidism due to autoimmune condition only requiring hormone replacement, psoriasis not requiring systemic treatment, or conditions not expected to recur in the absence of an external trigger are permitted to enroll. * History of organ transplant that requires use of immunosuppressives. * Active or prior documented inflammatory bowel disease (e.g. Crohn?s disease, ulcerative colitis) or a history of primary immunodeficiency. * Known history of tuberculosis. * Unresolved toxicities from prior anticancer therapy, defined as having not resolved to National Cancer Institute?s (NCI?s) Common Terminology Criteria for Adverse Events (CTCAE) (NCI CTCAE version \[v\]4.03) grade 0 or 1, or to levels dictated in the inclusion/exclusion criteria with the exception of alopecia. Subjects with irreversible toxicity that is not reasonably expected to be exacerbated by nivolumab may be included (eg, hearing loss) after consultation with the Bristol-Myers Squibb (BMS) medical monitor. * Subjects who active hepatitis B or C, or human immunodeficiency virus (HIV). * Receipt of live attenuated vaccination within 30 days of receiving nivolumab or anti-LAG3 antibody. * Prisoners or subjects who are involuntarily incarcerated. * Subjects who are compulsorily detained for treatment of either a psychiatric or physical (e.g, infectious disease) illness. * Uncontrolled or significant cardiovascular disease including, but not limited to, any of the following: * Myocardial infarction (MI) or stroke/transient ischemic attack (TIA) within the 6 months prior to consent. * Uncontrolled angina within the 3 months prior to consent. * Any history of clinically significant arrhythmias (such as ventricular tachycardia, ventricular fibrillation, or torsades de pointes, or poorly controlled atrial fibrillation). * Corrected QT (QTc) prolongation \> 480 msec. * History of other clinically significant cardiovascular disease (i.e., cardiomyopathy, congestive heart failure with New York Heart Association \[NYHA\] functional classification III-IV, pericarditis, significant pericardial effusion, significant coronary stent occlusion, poorly controlled venous thrombus). * Cardiovascular disease-related requirement for daily supplemental oxygen. * History of two or more MIs or two or more coronary revascularization procedures. * Subjects with history of myocarditis, regardless of etiology. * Subjects with history of life-threatening toxicity related to prior immune therapy (eg. anti-cytotoxic T-lymphocyte-associated protein \[CTLA\]-4 or anti-PD-1/PD-L1 treatment or any other antibody or drug specifically targeting T-cell co-stimulation or immune checkpoint pathways) except those that are unlikely to re-occur with standard countermeasures (eg, hormone replacement after endocrinopathy). * Troponin T (TnT) or I (TnI) \> 2 x institutional ULN. Subjects with TnT or TnI levels between \> 1 to 2 x ULN will be permitted if repeat levels within 24 hours are 1 x ULN. If TnT or TnI levels are \> 1 to 2 x ULN within 24 hours, the subject may undergo a cardiac evaluation and be considered for treatment, following a discussion with the BMS medical monitor or designee. When repeat levels within 24 hours are not available, a repeat test should be conducted as soon as possible. If TnT or TnI repeat levels beyond 24 hours are \< 2 x ULN, the subject may undergo a cardiac evaluation and be considered for treatment, following a discussion with the BMS Medical Monitor or designee.