Validity and limitations of the equipartition theorem

The Classical equipartition theorem can be resumed by the following statement

In equilibrium, nature seeks the equipartition of energy, in which each degree of freedom contains an average amount of energy $K_b T/2$

At high temperatures, this result is more or less accurate. But to low tempertarues, the typical examples that attack this claim were the ultraviolet catastrophe and the specific heat of diatomical gases and solids. They are quantum in nature and attack the principle rightly at the root, since not all degrees of freedom are quantized equally or can absorb the same amount of energy, so that the energy that a few degrees of freedom reject, is absorbed by another degree of freedom.

But are there more ways to overthrow this outcome? Wikipedia even talks about Chaos and other things. Can you discuss and criticize, using examples, the equipartition statement?

Thanks in advance.