Study to Evaluate Erlotinib With or Without SNDX-275 (Entinostat) in the Treatment of Patients With Advanced NSCLC

Inclusion Criteria * Cytologically or histologically confirmed NSCLC of stage IIIb or IV * Received at least 1 but no more than 2 prior chemotherapy or chemoradiotherapy regimens for advanced NSCLC (that did not include erlotinib and valproic acid) and progressed based on radiologic evidence * At least 1 measurable lesion by conventional or spiral computed tomography (CT) scan * Eastern Cooperative Oncology Group (ECOG) performance score of 0, 1, or 2 and life expectancy of at least 6 months * Paraffin-embedded tumor specimen available for correlative studies * Male or female over 18 years of age * Hemoglobin ≥ 9.0 g/dL; platelets ≥ 100 x 10\^9/L; absolute neutrophil count (ANC) ≥ 1.5 x 10\^9/L without the use of hematopoietic growth factors * Bilirubin and creatinine less than 2 times the upper limit of normal for the institution * Albumin ≥ 2.5 g/dL * Aspartate aminotransferase (AST) and alanine aminotransferase (ALT) less than 3 times the upper limit of normal for the institution * Prothrombin time less than 1.5 times the upper limit of normal for the institution * Potassium, magnesium and phosphorus within the normal range for the institution (supplementation is permissible) * Willing to use accepted and effective methods of contraception during the study (both men and women as appropriate) and for 3 months after the last dose of SNDX-275 * Patient or legally acceptable representative has granted written informed consent before any study-specific procedure (including special screening tests) are performed Exclusion Criteria * Prior stem cell transplant * Clinical evidence of central nervous system (CNS) involvement * Prior treatment with an histone deacetylase (HDAC) inhibitor or an epidermal growth factor receptor (EGFR) inhibitor * Currently taking known inhibitors of CYPA4, including but not limited to atazanavir, clarithromycin, indinavir, itraconazole, nefazodone, nelfinavir, ritonavir, saquinavir, telithromycin, troleandomycin, ≥ 10 mg prednisone, and voriconazole * Currently taking medication(s) on the prohibited medication list * Prior exposure to SNDX-275 * Systemic chemotherapy, radiotherapy, or treatment with an investigational agent without recovery to at least grade 1 or baseline before study drug administration * Daily treatment with ≥ 10 mg prednisone within 28 days before study drug administration * Local or whole brain palliative radiotherapy within 14 days before study drug administration * Currently active second malignancy, or any malignancy within the last 5 years other than cured basal or squamous cell skin carcinoma, cervical carcinoma in situ, carcinoma in situ of the bladder, or papillary thyroid cancer * Inability to swallow oral medications or a gastrointestinal malabsorption condition * Acute infection requiring intravenous (IV) antibiotics, antivirals, or antifungals within 14 days before study drug administration * Known human immunodeficiency virus (HIV) infection, or active hepatitis B or C infection * Another serious or uncontrolled medical condition within 90 days before study drug administration such as acute myocardial infarction, angina, ventricular arrhythmias, hypertension, diabetes mellitus, or renal or hepatic insufficiency * Known hypersensitivity to benzamides * Women who are currently pregnant or breast-feeding * Patient currently is enrolled in (or completed within 28 days before study drug administration) another investigational drug study * Patient has any kind of medical, psychiatric, or behavioral disorder that places the patient at increased risk for study participation or compromises the ability of the patient to give written informed consent and/or to comply with study procedures and requirements