Floppy trunk syndrome (abbreviated FTS, also known as flaccid trunk paralysis) is a condition that causes trunk paralysis in African bush elephants. Initially observed in 1989, the syndrome primarily affected bull elephants in several select regions in Zimbabwe."
Source : Wikipedia
The ailment can lead to death by malnutrition, as elephants use their prehensile trunks to forage for food. Affected animals sometimes develop coping strategies such as resting the affected trunk on the tusks.
The cause is presently unknown, and there is no treatment available. Theories of its origin include damage due to eating poisonous plants e.g. Loxodonta africana, Heliotropium ovalifolium and Blumea gariepina. (See this thesis from Alain Albert Guntern at the Université de Lausanne ( alternative link )) and also heavy metal poisoning from contaminated drinking water etc.