In America, we believe that anyone can grow up to be anything.

You want to be president? Go for it. Among recent presidents, Barack Obama, Bill Clinton, Lyndon Johnson and Dwight Eisenhower all came from modest backgrounds. Of course, not everyone will be president, which is why, for most people, one litmus test for how well America works is whether they rise above their parents economically.

Guess what? The long-term trends here are surprisingly good. In the shadow of the Great Recession, that’s a refreshing contrast to the prevailing gloom. A new report from the Pew Mobility Project, a nonpartisan research group, compares the incomes of forty-somethings now with their parents at a similar age in the late 1960s and early 1970s. Here are its generally upbeat findings:

● Most Americans (84 percent) exceed their parents’ income at a similar stage. Income gains were sizable across the economic spectrum. For the richest fifth, median income grew 126 percent from the late 1960s to the early 2000s, from $49,075 to $111,115. Among the poorest fifth, the median rose 74 percent, from $11,064 to $19,202. (All figures are in inflation-adjusted 2008 dollars.)

● Among sons, 59 percent had higher inflation-adjusted wages and salaries than their fathers. For the poorest fifth, 85 percent exceeded their fathers’ earnings. Higher family incomes also reflect more women with paid jobs. (From 1970 to 2000, the labor force participation rate of women 16 and over went from 43 percent to 60 percent.) Men’s earnings represent 61 percent of family income, down from 75 percent in the late 1960s.

● Along with higher incomes, there was much movement across class lines. Fully 60 percent of children born to the richest fifth of Americans in the late 1960s fell out of that category — 23 percent to the second-richest fifth and the rest scattered; 8 percent landed in the poorest fifth. As for upward mobility, about 57 percent of children born to the poorest fifth of Americans in the late 1960s moved up — 27 percent into the second-poorest fifth and 4 percent into the richest fifth.

Some findings are less comforting. For example, the wealth (homes, stocks, bonds) of the poorest 60 percent of Americans has decreased since the late 1960s. For the middle fifth, the decline was 5 percent; for the poorest fifth, the fall was a staggering 63 percent. Debt seems the most likely cause: Americans own more but also owe more. Greater borrowing reduces net worth.

More disturbing, black families remain concentrated at the bottom of the economic ladder despite large income gains: 49 percent of black families are in the poorest fifth, compared with 15 percent of whites.

But the main message is reassuring: A rising tide really can lift all boats.

Curiously, some commentators are arguing that the report shows the United States has too little mobility. Even Pew says its study reveals “a glass half empty.” It cites the facts that 43 percent of children born of parents in the poorest fifth remain in the poorest fifth and that 40 percent of children born in the richest fifth stay in the richest fifth. Parents are fate, it seems.

To understand this debate, you need to distinguish between two types of mobility — first, “absolute” mobility, comparing children’s incomes with their parents’; and second, “relative” mobility, comparing children’s position in the income distribution (often measured from the richest to the poorest fifth of Americans) with that of their parents.

As we’ve seen, most middle-aged Americans have achieved upward mobility by the first definition, at least before the Great Recession. Regarding relative mobility, for everyone who moves up in position, someone else must move down. One hundred percent of the people can’t be in the richest 20 percent.

A rigid class system — with everyone staying in their parents’ position — would be undesirable. It would discriminate against effort and talent, stifle ambition and discourage parents from helping their children. We don’t have that. Already, Pew reports, about 70 percent of children don’t end up in their parents’ position on the economic ladder: About 35 percent rise, 35 percent fall. Would we be better off if 50 percent rose and 50 percent fell? Doubtful.

Of course, parents matter. The family is society’s crucial institution for shaping character, behavior and economic opportunity. It’s better to be born to a richer rather than a poorer family; a two-parent family further improves the odds. But as anyone with children knows, even parents’ influence is limited. And if parents are limited, government interventions to improve children’s prospects are more so.

The real questions posed by the Pew study are unanswerable. What will govern the future? Will today’s young Americans duplicate their parents’ broad gains in living standards — or will their prospects be permanently damaged by the Great Recession?