Walking Through Walls

Yes, it is sort-of-possible. The phenomenon is called "Quantum Tunneling" and is a result of the wave-particle duality of matter. But don't go banging your head against a wall just yet. Although theoretically it is possible for a human being to undergo this process, the probability of its occurrence is vanishingly small, which basically means that it will never happen. However, tunneling is a daily part of being a microscopic particle, such as an electron.

The "wall" that they pass through is not actually a physical, material barrier, but rather a "potential" barrier that is created by energy. In the familiar world of classical physics, an object can never surmount a barrier that it does not have enough energy to surmount - this seems painfully obvious. For example, try as you might, you cannot throw a baseball into outer space because the force of gravity is too great compared with the energy you endowed the ball with by throwing it.  Scaling down to the atomic level, an example of a potential barrier is that created by nuclear forces to keep protons and neutrons inside a nucleus. By classical thinking, you could never find a proton or neutron outside the nucleus because they do not have enough energy to pass through the nuclear barrier. But reality proves otherwise. The wave properties of matter and the Uncertainty Principle grant microscopic particles a nonzero probability, quantified by a "transmission coefficient", of being transmitted through a barrier that requires more energy to pass through than the particle actually has (i.e the barrier has more potential energy than the particle has kinetic energy).

Quantum tunneling is exploited in many modern devices that we use everyday, such as in semiconductors used to power electronic devices. We also owe our sunshine to quantum tunneling! Inside of stars, individual protons combine in the process of nuclear fusion, releasing energy in the form of light. Classically, the temperature inside of stars would not give protons enough kinetic energy to overcome the electric potential force that repulses them (protons are positively charged, and like charges repel each other) in order to fuse together. But, through tunneling, a certain percentage of the protons pass through the electronic barrier, fuse together, and release light.