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It's an interesting question, but a bit vague. Even at room temperature, relatively needs to be considered for the motion of electrons.
You're probably thinking about bigger stuff though. The short answer is that temperature is unbounded so yes, there is a temp at which it is significant for the motion of all particles. I think inside of stars this can happen, but my knowledge jn that area is pretty limited.
Veritassium has a recent video about some of this that you may find interesting if you haven't already seen it.
Veritassium ignores a bunch of stuff in that video and hand-waves it away.
I only hear about his videos from other, better channels that correct his mistakes. He's dead to me ever since that "faster than light" electricity video where he didn't once use the word induction and made it sound super mystical. Fuck that guy with a thousand meters of wire.
Here's the video I saw on it. Anyone watching the Veritassium video should watch this after:
https://youtu.be/8Ks680LaL-Q
Or better yet, find a different video on the relativistic movement of electrons and electron holes in wires, and how it causes magnetism. I don't have one handy.
It's a really bad sign when half of his videos need corrections by other channels. Sure, you could say they're just riding on his popularity, but the fact that he needs corrections is the problem.
The video you linked summarizes the intent and benefit of Veritasium videos at about the 2:25 mark, stating that they are for a general audience. I agree that Veritasium isn't perfect, and doesn't provide complete depth, but they do a good job of creating interest in topics. So they accomplish their goal.
Additionally, the video you linked is wrong about the principles it discusses. The drift and diffusion velocity (group velocity) of electrons and holes is small compared to the speed of light. The relativistic effects discussed are caused by the phase velocity, which will be closer to the speed of light in the medium for even small currents.
Edit: originally, I incorrectly worded the last sentence which implied that the electrons and holes had a phase velocity equal to the speed of light. I hope the statement is more clear now, but I'm happy to provide additional clarification if necessary.