Standing humans aren't particularly stable. As most martial arts practitioners know, it doesn't take much to knock a standing person down. That's because, when standing, your center of gravity is above the point where you touch the ground and your feet provide a relatively small platform. If you tip even slightly, your center of gravity is no longer over your feet, and you either have to take a step or fall over.
If we aren't stable standing still, you might imagine that walking is that much more complex. But a paper in Physical Review Letters back in 1998 showed that some things that can't stand still at all can walk just fine.
To demonstrate this, Cornell researchers Michael Coleman (now at the University of Vermont) and Andy Ruina built a few tinkertoy machines that blithely toddle down hill, even though they would fall over instantly if you tried to stand them up.
The devices show that walking is possible without resorting to complex sensors or stabilizing mechanisms like gyroscopes. Although they are not very similar to humans, they suggest that human walking may have more to do with our mechanical structure than your brain or nervous system. Indeed, another Cornel project from 2001 resulted in a device that walks down a slight slope with a very human-like gate.
This research is among the first studies to suggest that that standing still for humans may be more difficult than walking because of the way our legs and bodies are built. So if there really are people who can't walk and chew gum at the same time, they would be in serious trouble trying to stand still and chew gum, I imagine.