The purpose of this research study is to determine if the combination of docetaxel, cisplatin, irinotecan and bevacizumab will help shrink metastatic esophageal or gastric cancer and how the cancer responds to this combination. Bevacizumab is a new drug that is believed to stop the formation of new blood vessels that carry nutrients to tumors. Bevacizumab is approved for use in metastatic colon and rectal cancer. Docetaxel, cisplatin and irinotecan are traditional chemotherapy agents that have been tested together in another clinical trial for esophageal and gastric cancer. It is hoped that adding bevacizumab to this regimen will make the treatment more effective.