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Subsumedpat It is illegal in some states to change your registration just to mess with the other party, although I don't know if anyone has ever been prosecuted for this. I do know that in 2008, Rush Limbaugh urged his listeners to register as Democrats and vote for Hillary in their primary in order to drag out the conflict between her and Obama as long as possible. It wasn't necessary for them to vote in the Republican primary as McCain was the sure winner by then. I don't know if anyone actually did this.
I have heard of people crossing party lines for this reason, especially in states where you can just ask for whatever party ballot you want. But I don't know if it's ever had an effect.