Negative temperatures occur when the derivative of entropy with respect to a system's energy is negative. In other words, the entropy decreases with added energy. As Daniel Schroeder points out in his Introduction to Thermal Physics, this may only occur when the total energy that a system may take is limited, such as a two-state paramagnet. In other, more common systems, such as a gas in a container, the total energy that the system may absorb is practically unlimited. This is why negative temperatures are not observed.
I think a lot of confusion in learning thermodynamics is that the common sense notion of temperature is very different from its thermodynamic definition. Other ideas, such as force or work, do not contradict common sense quite so much and are more readily adopted.
Finally, Schroeder references this article for an experiment in which negative temperature was observed.
A "lab notebook" about my life in science, programming, and philosophy.
