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Say I have 2 copies of Universal Network Access (Or any star card) in my deck, and I use Unleash Potential to bring a third copy into my hand. Fast forward a bit, and I manage to play Unleash Potential again. Could I use that to bring a fourth copy of that card into the game or does the rule of three still apply?

The rule only applies when building your deck. In this scenario you could bring in another copy.
Source: Wizards of the Coast (2019-02-22)    【Link to Ruling】
Tags: Unleash Potential