The Klaitman-Smalls' considerable investment in their children is becoming the norm for families like theirs who are in the top tiers of the country's income distribution. The resources the affluent are pouring into their children are also driving a growing divide between academic outcomes of the children of the well-to-do and those of everyone else's kids. That widening academic divide means that kids who are born poor and kids who are born rich are increasingly likely to stay that way once they reach adulthood.

When Martin Luther King Jr. gave his "I Have a Dream" speech a half-century ago, on Aug. 28, 1963, black children lagged behind their white peers in school by more than three years. For poor children, the picture was somewhat more encouraging: Those in the 10th percentile of income fell behind the children in the upper echelon of wealth by about a year or so. Poverty was a major obstacle, but not so large that it couldn't be scaled by the brightest and most ambitious.

Fifty years later, social class has become the main gateway--and barrier--to opportunity in America.

The country is far from fulfilling King's dream that race no longer limit children's opportunities, but how much income their parents earn is more and more influential. According to a 2011 research study by Stanford sociologist Sean Reardon, the test-score gap between the children of the poor (in the 10th percentile of income) and the children of the wealthy (in the 90th percentile) has expanded by as much as 40 percent and is now more than 50 percent larger than the black-white achievement gap--a reversal of the trend 50 years ago. Underprivileged children now languish at achievement levels that are close to four years behind their wealthy peers.

These days, middle-class children are also falling further behind their affluent peers. The test-score gap between middle-income (the 50th percentile of income) and poor children has remained stagnant; it's the gap between the top earners and the rest that is growing rapidly. And though more poor and middle-income children are completing college these days, they can't keep up with the growth in college graduates among the wealthiest families. A 2012 study by Reardon also found that "more and more seats in highly selective schools have been occupied by students from high-income families."

"Income has become a much stronger predictor of how well kids do in school," Reardon says. "Race is about as good a predictor as it was 30 years ago. It's more that income has gotten more important, not that race has gotten less important."

Jessica Klaitman's husband works in the finance industry, and she's a social worker who has worked part-time running support groups for new parents and teaching yoga. They do well enough to pay $4,400 a month in rent for a two-bedroom apartment in Brooklyn Heights, one of the borough's most-coveted neighborhoods, to pay for a full-time nanny in addition to their spending on preschool, and to take the family on regular vacations to see relatives out of state.