Lenalidomide and Cetuximab in Treating Patients With Advanced Colorectal Cancer or Head and Neck Cancer

Inclusion Criteria: * Patients must have histologically or cytologically confirmed malignancy that is metastatic or unresectable and for which standard curative or palliative measures do not exist or are no longer effective; eligible malignancies include: colorectal cancer KRAS wild-type and squamous cell head and neck cancer * No curative intent therapy available; there is no limit on prior number of therapies; prior epidermal growth factor receptor (EGFR)-directed therapy (tyrosine kinase inhibitors and monoclonal antibodies - including cetuximab, panitumumab, or investigational EGFR directed monoclonal antibodies) will be allowed in the phase I dose escalation; patients who have received monoclonal antibody therapy must be off all monoclonal antibodies four weeks (28 days) prior to study treatment; no chemotherapy within 28 days of trial medication * There will be an expansion cohort for colorectal cancer patients only; for the expansion cohort, there is no limit on prior chemotherapy; the colorectal expansion cohort will include patients with cetuximab or panitumumab-resistant or refractory disease (progression during cetuximab/panitumumab therapy or within 3 months of cetuximab/panitumumab therapy * Eastern Cooperative Oncology Group (ECOG) performance status =\< 2 (Karnofsky \> 60%) * Life expectancy of greater than 3 months * Leukocytes \> 3,000/mcL * Absolute neutrophil count \> 1,500/mcL * Platelets \> 100,000/mcL * Total bilirubin within normal institutional limits * Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) \< 2.5 x institutional upper limit of normal * Creatinine clearance \> 60 mL/min/1.73 m\^2 as calculated using modified Cockcroft-Gault formula * Females of childbearing potential (FCBP) must have a negative serum or urine pregnancy test with a sensitivity of at least 25 mIU/mL within 10 - 14 days and again within 24 hours prior to starting cycle 1 of lenalidomide; further, they must either commit to continued abstinence from heterosexual intercourse or begin TWO acceptable methods of birth control: one highly effective method and one additional effective method AT THE SAME TIME, at least 28 days before starting lenalidomide; FCBP must also agree to ongoing pregnancy testing; men must agree to use a latex condom during sexual contact with a FCBP, even if they have had a successful vasectomy; a FCBP is a sexually mature woman who: 1) has not undergone a hysterectomy or bilateral oophorectomy; or 2) has not been naturally postmenopausal for at least 24 consecutive months (i.e., has had menses at any time in the preceding 24 consecutive months); all patients must be counseled by a trained counselor every 28 days about pregnancy precautions and risks of fetal exposure * Men must agree to use a latex condom during sexual contact with a FCBP even if they have had a successful vasectomy * All patients must be counseled at a minimum of every 28 days about pregnancy precautions and risks of fetal exposure * Able to take aspirin (81 or 325 mg) daily for prophylactic anticoagulation (patients intolerant to aspirin may use warfarin or low molecular weight heparin) Exclusion Criteria: * Patients who have had chemotherapy or radiotherapy within 4 weeks (6 weeks for nitrosoureas or mitomycin C) prior to entering the study or those who have not recovered from adverse events (\>= grade 3) due to agents administered more than 4 weeks earlier * Patients may not be receiving any other investigational agents * Uncontrolled brain metastases; patients who have received definitive therapy, including radiation, and are not requiring ongoing medical therapy (i.e. steroids) for brain metastases will be allowed * History of allergic reactions attributed to compounds of similar chemical or biologic composition to lenalidomide or cetuximab or other agents used in study * Patients with a recent history of deep vein thrombosis (DVT)/pulmonary embolism (PE) requiring therapy (within 3 months) * Patients with history of toxicity \>= grade 3 with prior EGFR directed therapy * Patient with confirmed history of interstitial lung disease * Uncontrolled concurrent illness including, but not limited to, ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, or psychiatric illness/social situations that would limit compliance with study requirements * Pregnant women are excluded from this study; breastfeeding should be discontinued if the mother is treated with either agent * Human immunodeficiency virus (HIV)-positive patients on combination antiretroviral therapy are ineligible