I think there is still an overwhelming belief in American society, despite how far we’ve come in gender equality (19th amendment, women’s suffrage, is less than a hundred years old), that the male is the breadwinner and the female is the homemaker – which leads to some consternation when the female out-earns the male. Despite the gender neutral title, I think this question is more acute if you’re in a heterosexual relationship and you’re a guy who comes from a more fairly traditional family with a single breadwinner, your father. (I don’t know the dynamics of homosexual relationships, but it would be nice if someone from Queer Cents  could share their two cents on this)
So, if you’re the guy and your girlfriend, fiancée, or wife made more money than you, would you be bothered by it? If you’re the girl and your boyfriend, fiancé or husband made less money than you, would you be bothered by it, insomuch that you’d be afraid of making him feel bad?
I’d say that in my situation, I’d be okay with it in part because I see it more like “us” and less like “me and her,” if that makes sense. One of my friends and his wife, I know that very soon one day the wife will out earn the husband. The husband is perfectly fine with that because he sees it as a team so there is never the consideration of “her” out earning “him,” it’s “they” out earning “old they.” I think that’s healthy but not necessarily the average reaction.
I also think it’s perfectly acceptable to be bothered by it for a variety of reasons. If you and your SO one day want to drop to one income and raise children, just so one of you can stay home with the kids, I do believe that the mother is better at raising children than the father (before you scream gender discrimination, just consider the biological advantages of mothers). If the mother out earns the father, it’s harder to replace the higher income and so there’s no doubt in my mind that the father feels pressure.