Faced with medically-established facts, can a sincere person persevere in their unbelief?

In 1971, a group of Lutheran theologians from East Germany published a manifesto containing some touching lines: 

"At Lourdes, Fatima and other Marian shrines, impartial critics find themselves in the presence of supernatural facts which are intimately connected with the Virgin Mary, either because of her Apparitions, or because of the miraculous graces petitioned for and obtained through her intercession. 

These facts defy natural explanation. To date, 1,200 healings at Lourdes have been recognized by doctors as scientifically inexplicable. But the Catholic Church has declared only 44 as miraculous. For 30 years, 11,000 doctors, regardless of religion or scientific opinion, have had free access to the Medical Records Office. A cure declared miraculous therefore possesses the greatest possible guarantee.”

Voyage et Mission issue #113 by FJE

Source: Brother Albert Pfleger's Recueil marial (1991)