Income inequality has skyrocketed in the states over the past 35 years. In response to that fact, it's often been argued that what really matters is America's robust level of social mobility. That claim has, in turn, spurred a line of research showing that the level of social mobility in the United States is, in fact, not very high. For male workers, income is highly correlated with paternal income, so income inequality simply replicates from one generation to the next.
I often read people characterizing that research as proving that economic mobility has declined during the period in question, but that's much less clear. What it proves is that rags-to-riches (and riches-to-rags) stories are rare. But as Ryan Avent writes, some recent research seems to suggest that there's just very little mobility in general. Gregory Clark has shown that in Sweden, aristocratic surnames are overrepresented among lawyers by a factor of six. That's a pattern of inequality that dates back to the 17th century.
Something that I think ought to be put on the agenda in terms of understanding this is immigration. When politicians tell stories of upward mobility, they often make reference to parents or grandparents who immigrated here from abroad. And it's clearly true that moving to the United States is an effective way for many people to raise their income. And at different points in time, moving to the United States has been an effective way to evade systems of discrimination or persecution. But it's very possible to be a "land of opportunity"—in the sense that moving here is a valuable opportunity for many people—without being a land where native-born residents with low-income, English-speaking parents have an especially good chance of working their way up the economic ladder. It's also important to remember race in this context. The "good old days" were generally a period of extreme stratification along racial lines, which continues to have economic implications today.