Romidepsin and Erlotinib Hydrochloride in Treating Patients With Stage III or Stage IV Non-Small Cell Lung Cancer

To be eligible for study participation, patients must fulfill all of the following criteria: * Histologically confirmed locally advanced or metastatic (stage IIIB pleural effusion or stage IV) NSCLC; * Age ≥ 18 years; * Written informed consent; * Measurable disease as defined by Response Evaluation Criteria in Solid Tumors (RECIST); * ECOG (Eastern Cooperative Oncology Group ) performance status 0 to 1; * Serum potassium and magnesium greater than or equal to the lower limit of institutional normal range (electrolyte abnormalities may be corrected with supplementation to meet inclusion criteria); * Negative urine or serum pregnancy test on females of childbearing potential; * All women of childbearing potential must use an effective barrier method of contraception (an intrauterine contraceptive device \[IUCD\] or double barrier method using condoms or a diaphragm plus spermicide) during the treatment period and for at least 1 month thereafter. Male patients should use a barrier method of contraception during the treatment period and for at least 1 month thereafter. Hormonal methods of contraception such as the contraceptive pill or patch (particularly those containing ethinyl-estradiol) should be avoided due to a potential drug interaction (see Appendix D). * Adequate bone marrow, liver, and renal function as evidenced by * Hemoglobin ≥10 g/dL (transfusions and/or erythropoietin-stimulating agents are permitted) * Absolute neutrophil count (ANC) ≥1.5 x 109 cells/L • Platelet count ≥100 x 109 cells/L * Total bilirubin \<1.5 x upper limit of normal (ULN) * (Aspartate amino transferase) AST/SGOT(serum glutamic-oxaloacetic transaminase) and (amino alanine transferase) ALT/SGPT (serum glutamic-pyruvic transaminase) \<2.0 x upper limit of normal (ULN) or \<3.0 x ULN in the presence of demonstrable liver metastases * Serum creatinine \<2.0 x ULN * Clinically stable brain metastases are permitted Phase I study: * Prior erlotinib therapy is permitted (with a 3-week washout period) * Patients may have received prior anti-cancer therapy (with a 3-week washout period) or, at the discretion of the investigator, may be treatment-naïve Phase II study: * Patients must have received at least one and no more than two prior chemotherapy regimens for their advanced NSCLC * Patients may not have received prior erlotinib Patients are ineligible for entry if any of the following criteria are met: * Chemotherapy for NSCLC within 3 weeks prior to study entry; * Concomitant use of any other anti-cancer therapy; * Concomitant use of any investigational agent; * Use of any investigational agent within 4 weeks prior to study entry; * Any known cardiac abnormalities such as: * Congenital long QT syndrome; * QTc interval (corrected QT interval) Myocardial infarction within 12 months prior to study entry; * Other significant ECG abnormalities including type II second-degree atrio ventricular (AV) block, third-degree AV block, or bradycardia (ventricular rate \< 50 beats/min); o A history of coronary artery disease (CAD); eg, angina Canadian Class II-IV. In any patient in whom there is doubt, the patient should have a stress imaging study and, if abnormal, angiography to define whether or not CAD is present; * An ECG recorded at screening showing significant ST depression (ST depression of ≥2 mm, measured from isoelectric line to the ST segment at a point 60 msec from the end of the QRS complex). If there is any doubt, the patient should have a stress imaging study and, if abnormal, angiography to define whether or not CAD is present; * Congestive heart failure (CHF) that meets New York Heart Association (NYHA) Class II to IV definitions and/or ejection fraction \< 40% by MUGA ( multiple gated acquisition) scan or \<50% by echocardiogram and/or MRI; * A known history of sustained ventricular tachycardia (VT), ventricular fibrillation (VF), Torsades de Pointes, or cardiac arrest unless currently addressed with an automatic implantable cardiac defibrillator (AICD); * Hypertrophic cardiomyopathy or restrictive cardiomyopathy from prior treatment or other causes (if there is any doubt, see ejection fraction criteria above); * Uncontrolled hypertension (defined as blood pressure \[BP\] ≥160/95; or * Any cardiac arrhythmia requiring anti-arrhythmic medication; * Serum potassium or serum magnesium below lower limit of institutional normal range (electrolyte abnormalities may be corrected with supplementation to meet inclusion criteria) * Concomitant use of drugs that may cause a prolongation of the QTc interval . * Concomitant use of CYP3A4 inhibitors * Concomitant use of warfarin (due to a potential drug interaction); * Clinically significant active infection (including known infection with human immunodeficiency virus \[HIV\], hepatitis B, or hepatitis C); l \>480 milliseconds (msec); * Major surgery or radiation within 2 weeks prior to study entry; * Patients who are pregnant or breast-feeding; * Any significant medical or psychiatric condition that might prevent the patient from complying with all study procedures; * Prior exposure to romidepsin