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Polygons shifting position in PS1 games

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So, this has bugged me for a while, and I'm posting the question here because I have a feeling it has to do with floating point precision on older consoles (like the PS1) and older PCs. Perhaps the best way to explain the issue is to show it so, watch the following video and look at the table in the video at about 5:11.
Note how the corner vertices seem to bounce around from their original location. Since the model is static and the geometry doesn't change, I have to assume that it's an artifact of the rendering technique.

So, my question is, why? Why did this happen in the older games? And how was it fixed in later games? If someone can explain this in detail, I would appreciate that very much.



Edited: The effect can also be seen on the protagonist's right arm at about 7:03. Edited by Xerodesign

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