When Karen Coveler and her husband began trying to have a child, she told her obstetrician that she wanted to take all the DNA tests she could to determine whether she was at risk of passing on a genetic disease to her child. Based on her Ashkenazi Jewish background, Ms. Coveler was offered 10 tests, all of which were negative, and went on to have a normal pregnancy. It was not until her son, Benjamin, was born that she discovered he was deaf. And it was not until a few weeks later that she learned a simple blood test could...