Can a container of cold water have the same thermal energy as a container of hot water? Why or why not?

Oooh, good question. The simple answer is, "Yes", but there are a lot of assumptions in that answer.

Thermal energy is the measure of heat added to or removed from a substance. If we assume the amount of energy (heat) added to each container is the same, but the comtainers of water are of significantly different volumes, then the larger container will remain cold, while the smaller container becomes hot. Basically, it's quicker to boil a small amount of water compared to a large volume of water.

However, if the containers are of the same or similar volumes, then it is nearly impossible for one to have different thermal energy than the other.

I say "nearly impossible", because we can imagine situations where the containers are made of materials that have vastly different heat conducting properties, therefore, one may conduct heat to the environment much faster than the other, causing its water to lose thermal energy and remain cold.