What exactly was Vincent van Gogh’s problem? This summer, in a Dutch exhibition and a related symposium, art historians and physicians will debate a question that has long puzzled the medical and art worlds.

“On the Verge of Insanity: Van Gogh and His Illness” opens July 15 at Amsterdam’s Van Gogh Museum and runs until Sept. 25. The 60 artworks and objects on view will include a payment receipt for an asylum stay by the artist and a February 1889 petition from citizens of Arles, the southern French town where van Gogh had...