Directors of the local centers have managed this caseload with the tools they have, in part by seeing to it that would-be clients get educational materials or online resources right away, to give them something to study while they wait for an appointment.

Sarah Norman, 45, a pediatric nurse who sought help from the Oxford center last year for depression, said she was referred to group therapy because the waiting list for individual therapy was so long. When the group therapy ended after four sessions, she remembered: “I was a bit frustrated. I thought I could have used a couple more sessions.”

She did, in time, improve, and is very grateful for the treatment. The same cannot be said with any certainty about the 40 percent of people who the data show were lost to the program after the initial assessment phone call. About two-thirds of them were not depressed or anxious enough to qualify for the therapy, or decided it wasn’t for them, Dr. Clark’s data shows.

That leaves about 125,000 men and women who may have needed help but didn’t get it. “These are people we’d like to reach, and we are pushing the services hard to do that,” Dr. Clark said.

Expanding the Reach

John Pimm, the psychologist who directs the Buckinghamshire center, found in 2013 that he could strikingly raise recovery rates by having his therapists give some patients two or three extra sessions; and by extending call times for phone therapy, working more carefully at the critical junctures, when people begin, or end, therapy.

“We have created this program, and now we are playing catch-up,” Dr. Pimm said in an interview in his office in High Wycombe. “We cannot train therapists fast enough, and the low-intensity phone therapists turn over pretty quickly. We have to figure out how to keep them in the job longer.”