Enrollment at American colleges is sliding, but competition for spots at top universities is more cutthroat and anxiety-inducing than ever. In the just-completed admissions season, Stanford University accepted only 5 percent of applicants, a new low among the most prestigious schools, with the odds nearly as bad at its elite rivals.

Deluged by more applications than ever, the most selective colleges are, inevitably, rejecting a vast majority, including legions of students they once would have accepted. Admissions directors at these institutions say that most of the students they turn down are such strong candidates that many are indistinguishable from those who get in.

Isaac Madrid applied to 11 colleges, a scattershot approach that he said is fairly typical at his private high school, Bellarmine College Preparatory in San Jose, Calif. Students there are all too aware of the long odds against getting into any particular elite university. “It was a crazy amount of work and stress doing all those essays by the deadline and keeping up my schoolwork, and waiting on the responses, and we had more than $800 in application fees,” he said.

Mr. Madrid, 18, got a taste of how random the results can seem. He was among the 95 percent turned away by Stanford, but he got into Yale, which he plans to attend, and he admitted having no real insight into the reasons for either decision.