Nobody who worked on this film was safe from injury, but in the case of the winged monkeys, someone really should have seen this coming. Originally, they weren't going to be played by people at all. It had been decided that the animals would be applied onto the film via animation after the film had been shot. Somewhere along the line, though, it was decided that this would jar badly with the live action of the rest of the film, and so real people in costumes should be used instead.
Given the limitation of special effects in 1939, giving the illusion that anybody was actually flying without it looking clearly fake was a difficult task. Small actors in uncomfortable costumes were attached to piano wires and dangled above the haunted forest for the scene they were in. The wires weren't up to the task and several of them snapped, leaving the performers to plunge helplessly several feet to the floor.