A Study of Ramucirumab (LY3009806) or Necitumumab (LY3012211) Plus Osimertinib in Participants With Lung Cancer

Inclusion Criteria: * Have a diagnosis of NSCLC with at least 1 measurable lesion assessable using standard techniques by the Response Evaluation Criteria in Solid Tumors Version 1.1 (RECIST 1.1). * Have T790M-positive status using a test validated and performed locally after disease progression on EGFR tyrosine kinase inhibitor (TKI) treatment. * Have Eastern Cooperative Oncology Group Performance Status (ECOG PS) of 0 or 1 at the time of enrollment. * Have serum albumin that is ≥25 grams per liter at the time of enrollment. * Have adequate organ function, with all screening labs performed within 7 days of treatment initiation. * Have a life expectancy of ≥3 months. * Have resolution, except where otherwise stated in the inclusion criteria, of all clinically significant toxic effects of prior systemic cancer therapy, surgery, or radiotherapy to Grade ≤1 by National Cancer Institute Common Terminology Criteria for Adverse Events (CTCAE) Version 4.0. Exclusion Criteria: * Previous treatment with an EGFR monoclonal antibody (except for past treatment for squamous cell carcinoma of head and neck or metastatic colorectal cancer). * Previous treatment with osimertinib or third generation EGFR TKIs. * Participants with symptomatic or growing brain metastases less than 4 weeks prior to enrollment. * History of drug-induced interstitial lung disease (ILD), ILD, or radiation pneumonitis requiring treatment with steroid prior to study enrollment, or any evidence of clinically active ILD. * Have a significant bleeding disorder or vasculitis or had a Grade ≥3 bleeding episode within 12 weeks prior to enrollment. Participants with a history of gross hemoptysis (defined as bright red blood of ≥1/2 teaspoon) within 2 months prior to enrollment are excluded. * Have experienced any arterial thrombotic event or arterial thromboembolic event, including myocardial infarction, unstable angina (history or evidence of current clinically relevant coronary artery disease of current ≥Class III as defined by Canadian Cardiovascular Society Angina Grading Scale or congestive heart failure of current ≥Class III as defined by the New York Heart Association), cerebrovascular accident, or transient ischemic attack, within 6 months prior to enrollment. * Have a history of deep vein thrombosis, pulmonary embolism, or any other significant venous thromboembolism (venous catheter thrombosis or superficial venous thrombosis not considered "significant") during the 3 months prior to study enrollment. Participants with venous thromboembolism occurring 3 to 6 months prior to study enrollment are allowed, if being treated with low molecular weight heparin. * Have a history of gastrointestinal perforation and/or fistula within 6 months prior to enrollment. * Have a bowel obstruction, history or presence of inflammatory enteropathy or extensive intestinal resection (hemicolectomy or extensive small intestine resection with chronic diarrhea), Crohn's disease, ulcerative colitis, or chronic diarrhea. * Have uncontrolled hypertension, as defined in CTCAE Version 4.0, prior to initiating study treatment, despite antihypertensive intervention. CTCAE Version 4.0 defines uncontrolled hypertension as Grade \>2 hypertension; clinically, the participant continues to experience elevated blood pressure (systolic \>160 millimeters of mercury \[mmHg\] and/or diastolic \>100 mmHg) despite medications. * Are receiving chronic therapy with any of the following medications within 7 days prior to enrollment: * nonsteroidal anti-inflammatory agents (NSAIDs; such as indomethacin, ibuprofen, naproxen, or similar agents). * other anti-platelet agents (such as clopidogrel, ticlopidine, dipyridamole, or anagrelide). * Have radiologically documented evidence of major blood vessel invasion or encasement by cancer. * Have radiographic evidence of pulmonary intratumor cavitation, regardless of tumor histology. * Are receiving concurrent treatment with other anticancer therapy, including other chemotherapy, immunotherapy, hormonal therapy, chemoembolization, or targeted therapy or radiotherapy treatment to more than 30% of the bone marrow or with a wide field of radiation within 4 weeks prior to enrollment. * Have abnormal cardiac findings. * Have undergone chest irradiation within 2 weeks prior to study drug administration, have not recovered from all radiation-related toxicities, or requires corticosteroids. A 2-week washout is permitted for focal palliative radiation to non-central nervous system disease.