Since at least the time of Abraham Lincoln, presidents have sent letters of condolence to the families of service members killed in action, whether the deaths came by hostile fire or in an accident. So after his son killed himself in Iraq in June, Gregg Keesling expected that his family would receive a letter from President Obama. What it got instead was a call from an Army official telling family members that they were not eligible because their son had committed suicide. “We were shocked,” said Mr. Keesling, 52, of Indianapolis. Under an unwritten policy that has existed at least...