h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

Interesting use of my coffee break!
Well, it's always been accepted as a classic quiz question that the elephant is the only mammal that cannot jump.
Whether it can run or not depends on your interpretation of the verb. Walking quickly can have the same effect if you can't escape from the charging beastie.
On the other [joky] interpretation, it depends on the animal's consumption of fibre and water. Prunes can make anyone run!

Researchers (including myself) have confirmed that elephants don't leave the ground with all 4 feet as this article states, but they do change from a typical walk (2-4 feet on the ground at all times, very stiff legs) to a different gait with often just 1 foot on the ground, and much more springy legs. In the modern biomechanical definition of running, this is what running is all about-- bouncing with pogo stick-like, bent legs, not vaulting over rigid straight legs.

They can do this up to 15mph (24kph). Speeds of 25 mph (40kph) attributed to them seem to be just tall tales (it is hard to accurately gage speed by eye or speedometer, especially if chased by an angry elephant). This applies to both African and Asian elephants of all ages and stories.

oops meant 'species' not 'stories' at the end there