If we take heat capacity to be defined as "the ratio of the heat added to the temperature rise":
$$ C=\frac{\text{d} Q_{rev}}{\text{d}\theta}$$
then this leads me to ask: can this ever be negative? That is to say, are there any materials which cool as you add heat to them?
There are certainly systems that have negative heat capacities, and in fact they come up all the time in astrophysics.
As a general rule, gravitationally bound systems have negative heat capacities. This is because in equilibrium (and remember we can't do classical thermodynamics without equilibrium anyway), some form of the virial theorem will apply. If the system has only kinetic energy $K$ and potential energy $U$, then the total energy is of course $E = K + U$, where $E < 0$ for bound systems. In virial equilibrium where the potential energy is purely gravitational, then we also have $K = -U/2$. As a result, $K = -E$, and so adding more energy results in a decrease in temperature.
Examples include stars and globular clusters. Imagine adding energy to such systems by heating up the particles in the star or giving the stars in a cluster more kinetic energy. The extra motion will work toward slightly unbinding the system, and everything will spread out. But since (negative) potential energy counts twice as much as kinetic energy in the energy budget, everything will be moving even slower in this new configuration once equilibrium is reattained.
At some level, this all comes down to what you're defining as temperature. Recall that temperature simply accounts for the flow of heat into whatever you've defined as your thermometer. If your thermometer couples to translational kinetic energy but not to gravitational potential energy, then you get the situation above.
I'll leave it to someone else to answer in terms of solid materials or inverted populations.
We don't need to go to astrophysics for this. In the reversible expansion of a plain vanilla ideal gas, if one does not add sufficient heat, the temperature will drop (and, by this definition, the heat capacity will be negative). This can happen any time work is done such that there is not enough heat added to increase the internal energy. This is why $dQ/d\theta$ is such a poor way of defining heat capacity. When defined this way, heat capacity is not even a physical property of the material. In classical thermodynamics, heat capacity is more properly defined in terms of the partial derivatives of internal energy and enthalpy with respect to temperature.
There are two different definitions of heat capacity, heat capacity at constant volume and heat capacity at constant pressure. The reversible expansion of an ideal gas cannot be done at constant volume. It cannot be done at constant pressure without adding heat.
Short answer is "no". The theory shows that heat capacities are positive. The negative heat capacities mentioned in the literature are based in misunderstandings of this theory.
For instance, the astrophysicists' argument uses the virial theorem to transform the sum of kinetic and potential energy $E=K+\Phi$ into $E= -K$ and then uses $K= \frac{3}{2} Nk_BT$ to get
$$C_V \stackrel{wrong}{=} \frac{dE}{dT} = -\frac{3}{2} Nk_B$$
which is a negative quantity, but is not the heat capacity of the system. The mistake is that the heat capacity $C_V$ is defined by a partial derivative at constant volume
$$C_V = \left( \frac{\partial E}{\partial T}\right)_V$$
The kinetic energy is a function of temperature, whereas the potential energy is a function of volume $E(T,V) = K(T) + \Phi(V)$, which means
$$C_V = \left( \frac{\partial E}{\partial T}\right)_V = \frac{3}{2} Nk_B$$
and we recover a positive heat capacity in agreement with both Schrödinger statistical mechanics theorem and with classical thermodynamic stability theory.
EDIT:
It seems my reply is being down-voted with invalid arguments such as "there is no confining volume", $(3/2)Nk_B$ is only for ideal gases, etcetera).
First, I am not considering the more general case, I am simply refutting the usual astrophysicists' argument and explaining (i) why they are not computing heat capacities, (ii) why finding regions where $(dE/dT) < 0$ does not identify systems with negative heat capacity, and (iii) why the statement "when heat is absorbed by a star, or star cluster, it will expand and cool down" is ignoring that volume is not held constant and therefore that $dQ \neq dE$.
My above answer used an abstract model, but we can be more specific. Consider the system discussed in [1]. The energy is $E = aT–b/r$, with $a = (3/2)Nk_B$ and $b$ an unspecified constant. Differentiating this energy and using the first numbered equation in [1], we obtain $dE = adT + PdV$ and a positive heat capacity for this gravitational system
$$C_V = \left( \frac{\partial E}{\partial T}\right)_V = a >0$$
D. Lynden-Bell and R. M. Lynden-Bell obtain the wrong result
$$C_V \stackrel{wrong}{=} \frac{dE}{dT} = -a$$
because they are not computing a heat capacity. Additional mistakes and physical inconsistencies in the arguments of LBLB are discussed in [2].
[1] Negative heat capacities do occur. Comment on “Critical analysis of negative heat capacities in nanoclusters” by Michaelian K. and Santamaría-Holek I. 2008: EPL 82(4), 43001. Lynden-Bell D.; Lynden- Bell, R. M.
[2] Reply to the Comment by D. Lynden-Bell and R. M. Lynden-Bell 2008: EPL 82(4), 43002. Michaelian K.; Santamaría-Holek I.
