Exactly right. In thermodynamics, the defining relationship for temperature 1/T = dS / dE, where T is temperature, S is entropy, and E is energy (and dS / dE means "the change in entropy as you add energy"). Now the entropy is, roughly speaking, "the number of possible states" of all the constituent atoms. So, positive temperature means that if you add energy to the system, there are more things each particle could randomly do (more states become available). This makes sense for systems with low total energy -- the atoms spend most of their time sitting around at low energy, and only occasionally get bumped around into higher energy states, so adding energy means "more possibilities" for each atom.
However, in a system where the number of high-energy states for each atom is limited, you can end up with the opposite situation -- if you add a lot of energy to the entire system, nearly all the atoms have to spend all their time in their own high-energy state, and only occasionally fall back down to their low-energy state. Adding even more energy to the system makes the situation even more restricted -- i.e. there are fewer possible "low energy" states available for each atom. So in this case dS / dE becomes negative -- more energy = fewer possible states. And so that means negative temperature.