When did "skipping" grades become common and when did it stop in USA?

As far as I know it still very rarely happens. I had several uncles who skipped a year, in high school. Not straight from primary school or first year high school to university. And that still happens too, though very rarely. At least in the Dutch educational system.

It is rare enough to hit the news when it does. I recall such a promotion happened last year or the year before. A student went from pre university (atheneum or gymnasium in the Dutch system) to university.

I can't give any links, but it does happen. Don't know about the US system, so I can't help you out there. It certainly wasn't common. My uncles skipped classes in the time period given (1910-1940), it happened more often on primary schools. Back then it wasn't unusual (but certainly not common) if a gifted child skipped a grade in (a Dutch) primary school.

I have a story idea about a schoolchild who, among other unusual atributes, skips grades, and I just found this:

http://theconversation.com/should-i-grade-skip-my-gifted-child-663591

Claiming that about 1 percent of school students in the USA skip grades, giving as its source:

http://www.accelerationinstitute.org/research/grade_acceleration_wells_lohman_marron.pdf2

Therefore, it seems that "skipping" grades has not been entirely phased out in the USA yet, despite the assumption in the OP.

Skipping grades and other forms of "meritocracy were almost brought to an end by Progressive Education that started in the late 19th century, and picked up steam in the 1920s. As late as the publication of Herman Wouk's "City Boy" (1928), it was commonplace, but fell out of favor shortly thereafter. In the early 1890s, an 11 year old immigrant, Mary Antin, was forced to start school in the first grade, but "skipped" to the sixth grade in one year. This would not happen nowadays.

"Traditional" education, which held sway in the 19th century in America, was a carrot and stick affair. ( won't go into the 18th century since there was no "USA" for most of the century. A student either knew things associated with a certain grade level, or s/he didn't. If not, the student was "held back." If "advanced" for their age, students were pushed forward into groups with older people. That meant that in any given grade, there could be people of several age levels. Prior to the beginning of the 20th century, before "universal" education, children would start school at different ages, and by that fact alone would reach a given grade at different ages, so no one thought anything of it then.

The theme of the "progressive" education was that the purpose of education was to help students progress in life, not in school, so that schooling was the means to an end, not an end of itself. This coincided in America with "universal" education. For a "life" educational project, it helps to have students of roughly the same age and experience, not academic ability. That meant that classes would be h*m*geneous by age, and heterogeneous by educational accomplishment (the reverse of traditional education). But the "progressive" educators had an answer for that; teachers were supposed to treat students as individuals and teach them at the "own" levels, not as members of a class. That's possible when you have small class sizes, typically no more than 25-30. The teacher's other function, equally important to education, was to conduct "group" activities.

So skipping grades went out of fashion when the idea of sorting students by age replaced the idea of sorting them by academic achievement. It became somewhat popular during and shortly after World War II, when society placed a premium on "efficiency" (even at the collegiate level, George Bush Sr. completed a four year program at Yale in two and half years). But "progressive" education regained its hold by the 1950s.

Additional source: Hyman Rickover, "American Education: A National Failure."