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Wells was born at Brambridge, Hampshire in 1536, and was christened with the name of the local saint and bishop Swithun. He was for many years a schoolmaster at Monkton Farleigh in Wiltshire. During this period, he attended Protestant services, but in 1583, was reconciled to the Catholic Church. In 1585 he went to London, where he purchased a house in Gray's Inn Lane.
In 1591, Edmund Gennings was saying Mass at Wells's house, when the priest-hunter Richard Topcliffe burst in with his officers. The congregation, not wishing the Mass to be interrupted, held the door and beat back the officers until the service was finished, after which they all surrendered peacefully. Wells was not present at the time, but his wife was, and she was arrested along with another priest by the name of Gennings, Polydore Plasden, and three laymen named John Mason, Sidney Hodgson, and Brian Lacey. Wells was immediately arrested and imprisoned on his return. At his trial, he said that he had not been present at the Mass, but wished he had been.
He was sentenced to die by hanging, and was executed outside his own house on 10 December 1591, just after St. Edmund Gennings. On the scaffold, he said to Topcliffe, "I pray God make you a Paul of a Saul, of a bloody persecutor one of the Catholic Church's children." His wife, Alice, was reprieved, and died in prison in 1602.