In the last four years, the school, led by a team of talented and ambitious young instructors, has seen its enrollment more than triple — there are around 170 students now, up from 50. But that success may ultimately jeopardize the school. The surge in students has led to increased anxiety about its relationship with Bronx Community College, which has allowed the school to operate out of its music department on Saturdays free of charge.

“We’re really at capacity now in this location,” said Douglas W. Meyer, the conservatory’s music director and treasurer. He described the program’s arrangement with the college for the last three decades as an essential part of its operating model. “We’re able to provide competitive compensation to our teachers and the lowest possible tuition rates to our students because we are the recipient of the largess of the college, for which we are eternally grateful,” he said.

“The good news is, we don’t have to maintain permanent facilities,” Mr. Meyer explained. “The flip side is that we’re here at the pleasure of the college.”

Changes to the arrangement , he said, could easily pose an existential threat; a rise in costs to offset rental payments, for example, could push the price of lessons beyond the reach of many students.