Columbia University’s elite primary school on 110th Street and Broadway has a popularity problem. With too many children competing for too few spots, Columbia has rejected dozens of students whose families thought they were assured of acceptance.

A surge in applicants to the School at Columbia University, as it’s known, has led to a tightening of admissions policies. One casualty is the unofficial practice of automatically allowing students’ siblings to enroll.

Parents outraged at the new policies gathered at the school last night at a PTA meeting to plead for a solution to a space problem that appears only to be getting worse. Parents are demanding emergency measures that would allow siblings to attend.

The issue of who is allowed to attend the school has long been a sore spot for Columbia.

Columbia initially conceived the School at Columbia more than four years ago as a powerful recruitment tool for attracting and keeping faculty members living in the Morningside Heights neighborhood. Faculty members receive a discount of 50% on the tuition of more than $20,000 a year.

Community leaders approved construction of the school only after intense negotiations with the university, which promised that half of the student body would be drawn from families living in Community School Districts 3 and 5 and not affiliated with Columbia. These so-called community students would be chosen by a lottery system, and many of them would receive need based financial aid.

When the school admitted its initial kindergarten-through-fourth-grade classes two years ago, half of the slots proved to be more than enough to accommodate interested families of Columbia faculty members and employees. As the school’s reputation for excellence spread, however, the number of professors wishing to enroll their children has grown considerably, to the point where Columbia is now rejecting some children of faculty members and other employees of the university.

The number of Columbia applications is twice the number of available spots, according to the president of the school’s PTA, Rafael Yuste, an associate professor at the university. For the first time, the school has put in place a lottery system for Columbia-affiliated children who are applying to enroll next fall.

Mr. Yuste said in a letter to parents that Columbia has placed a number of faculty children on a waiting list. “These families believed that their children would be admitted by right and they understandably did not make alternative plans, so the education of their children for next year is uncertain,” he wrote.

The School at Columbia occupies the first five floors of a 12-story building on West 110th. The school is adding a seventh grade next year, which will increase the student body to more than 400 students.

“Regardless of whether one agrees or disagrees with the new policy, the fact remains that we are facing a serious structural problem, since the School has quickly grown to be too small for the number of families requesting admission,” Mr. Yuste wrote.

With its five-to-one student-teacher ratio, its use of cutting-edge educational technology, such as computerized blackboards, and its generous financial-aid packages, the school has also accumulated a deficit, which has alarmed university officials. The New York Sun reported last summer that the deficit was at $9 million.

Columbia’s administration has said it built the school with an expectation that it would be continually subsidized. University officials have also said they have no intention of breaking promises they made to the community regarding the ratio of the student body.