Jori Johnson took the practice SAT test as a high-school student outside Chicago. Brochures later arrived from Vanderbilt, Stanford, Northwestern and the University of Chicago.

The universities’ solicitations piqued her interest, and she eventually applied. A few months later, she was rejected by those and three other schools that had sought her application, she said. The high-school valedictorian’s test scores, while strong by most standards, were well below those of most students admitted to the several schools that had...