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Why does wiggling some thin metal objects back and forth cause them to snap? For example, a paperclip. Is this a property shared across all metals, or only some?

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Bending a metal adds defects to the material.
As defects accumulate, the metal gets harder, stronger and more difficult to bend whilst at the same time getting hotter.
This is called work hardening.
However cracks also form leading to fatigue failure.

So try it with a metal (ductile material) paper clip and a plastic (brittle material) paper clip.
You should find that even trying to straighten out a plastic paper clip breaks it whereas you can straighten out and cycle a metal paper clip many times before it breaks.
Also note it getting harder to bend the metal paper clip before it breaks and it getting hotter.

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