American Dream is all about upward mobility. But it depends on how you define income mobility.
According to the recent research by San Francisco Fed, America is indeed very mobile when mobility is defined by absolute mobility: each generation achieves a higher economic status than the prior generation. But if mobility is measured by relative mobility, the extent to which individuals can change rank in the income distribution relative to their parents, America is much less mobile.
Two graphs to illustrate the story (graph courtesy of SF Fed):