Yes, so long as the temperature is high enough that the quantum discreteness of the energy levels can be ignored.
In classical statistical mechanics, it is possible to prove that any degree of freedom which contributes quadratically to the energy (i.e., kinetic energy, rotational energy, elastic potential energy) will contribute a factor of $\frac12 k T$ to the average energy of the system. The proof requires you to take into account the energy of every possible state of the system, and since we're thinking about things classically, this means that you have to integrate over all of the states of the system.
In quantum statistical mechanics, though, this integral is replaced by a discrete sum. If the quantum states are "close enough" in energy, then this sum is approximately equal to the integral you would perform in the classical case (think of taking a Riemann sum where the intervals are very narrow.) But if the quantum states are widely spaced, then the sum and the integral can give wildly different results (think of taking a Riemann sum where the intervals are very wide.)
The general rule for when the energy states are "close enough" is that the difference in the energy levels must be significantly smaller than the thermal energy scale $kT$. An example of this can be seen in the heat capacity for molecular hydrogen:
At low temperature, the spacing between the rotational energy states and the vibrational energy states is much greater than $kT$, and so H2 behaves like a monoatomic gas. At room temperature, $kT$ is smaller than the rotational spacing, but still larger than the vibrational spacing, and so the rotational modes contribute an extra $\frac{1}{2} kT$ each to the heat capacity. And at even higher temperatures, $kT$ is large enough that the vibrational modes start to "turn on" as well. Unfortunately, H2 dissociates into atomic hydrogen before these vibrational modes contribute their full $k T$ ($\frac{1}{2}kT$ for kinetic energy, $\frac{1}{2}kT$ for potential) to the heat capacity.
Another example of this is the ultraviolet catastrophe. In classical physics, a mode of the electromagnetic field in a cavity contributes quadratically to the energy (since the energy is proportional to the square of the amplitude of the field.) Thus, classically, the energy of each mode should be $\frac{1}{2} kT$, and since there are an infinite number of modes, there should be an infinite amount of electromagnetic energy in a cavity at any finite temperature. The paradox was resolved by Planck, by assuming (ad hoc) that the energy in these vibrational modes could only occur in discrete packets of energy $hf$; so the highest-frequency modes have a wide spacing compared to $kT$ and don't behave classically.