I've encountered one person in my life who seemed to be mysteriously healed through prayer. But I can't be 100 percent certain. I just know her recovery seemed unlikely, and very surprising. She had a brain tumor and had to use a walker due leg problems, and a Episcopalian canon at the local cathedral prayed for her and laid hands on her (I also prayed for her), and a few weeks later when I saw her, her walker was gone and she told me that the doctors told her, her brain tumor was gone also. But I lost contact with her and I wasn't that close to her to understand her story completely.
On the other hand, God hasn't really answered most of the prayer I've made. And reports of miraculous healing occasionally do happen in other religions. For instance, Arvol Looking Horse is a native American shaman who claims he was healed from a spinal injury by his dead grandmother. And a Taiwanese nun, Ven. Master Cheng Yen, claims as a child praying to Guan Yin healed her mother of a perforated stomach, which lead her to become a vegetarian.