Plenty of people with Irish roots can apply for an Irish passport. It is useful when travelling in Europe and you can work in any of the EU countries that are signed on to the Schengen Agreement.
In the case of Bachmann, her husband has allegedly applied for Swiss citizenship, which makes Michele eligible to apply for a Swiss passport. She can elect to do that if she wants. Presumably, she could renonunce any Swiss citizenship period. I am not familiar with Swiss nationality laws in that regard.
Personally, I would to see the US prohibit dual nationality similar to what the Germans do.