Does your faith group preach that some diseases can be miraculously cured?
Yes, we do. In theological language, a miracle is an extraordinary event, perceptible to the senses, produced by God in a religious context as a sign of the supernatural. By means of them, God alerts men and women to His salvific intentions. He also, at times, witnesses to and accredits those who claim to speak and act in His name, such as in all the miracles of Jesus. (The signs worked by Jesus invite belief in the Father who sent him and assure those who turn to him in faith, that he can grant what they ask, as stated in no. 548 of the Catechism.) There is also the example of the Apostles who worked wonders and signs, as outlined in the Acts of the Apostles (2:43). In these cases, the miracles testified to the witness the Church bore to the Kingdom of God.
The teachings of the Church concerning miracles can be traced back to the pronouncements of the first Vatican Council, when it declared that they show forth Gods omnipotence and infinite knowledge. Since miracles are, by definition, events perceptible to the external senses, there is no reason why they should not be ascertainable, either by eyewitnesses or others such as physicians.
There are many which have stood up to the rigid scrutiny of the canonization process. Some of the cures that have taken place at Lourdes, for example, are thoroughly medically documented. What do miracles teach us? More than anything, miracles have served to deepen our faith. In his encyclical Humani generis, Pope Pius XII best wrote what miracles mean, when he spoke of: "the many wonderful external signs God has given, which are sufficient to prove with certitude by the natural light of reason alone, the divine origin of the Christian religion".