The Pew Charitable Trust's new report on economic mobility contains data looking at "how closely tied a person’s place on the economic ladder is to that of his or her parents". The headline data from the report is that while many Americans are becoming wealthier than their parents, that the most significant economic gains are being channeled to the highest income brackets.
But I want to focus on a different chart, showing how likely you are to stay in the income quintile you are born into:

According to Pew, 40% of those born in the bottom stay in the bottom, 40% of those born into the stop stay at the top.
I like this study because it builds on research done in 2011 which looked at what factors encouraged downward mobility. When I spoke to Pew back then, they placed a lot of importance on educational attainment and whether individuals got married:
Regardless of how we measured downward mobility … we see the same pattern emerging. Post-secondary education and higher educational attainment matters ... This is a key reason why some people might be able to maintain their middle class status and some may not.
The second finding is that marital status is a key piece that explains why people stay in the middle class or fall out of it.
In this current generation of adults, we have seen a huge increase in two-earner family. It’s very important for family incomes that woman have entered the labor market.
When you look at people who become divorce, separated, or who are never married, it’s less surprising that they would have less of a chance of staying in the middle class than someone who is married.
The data Pew used for the latest study runs from 1968 up till 2008. (According to the report's appendix: "For the children, income is computed as the mean value of total family income taken in 2000, 2002, 2004, 2006, and 2008.") This means the data is from a pre-crash environment.
Undoubtedly in 10 or 20 years, when we start looking at the generation that comes of age in this economic climate, the results will be even worse as more young Americans put off marriage, forego completing their education, and otherwise fail to establish stable careers.