Eleven men, nine women were killed, officials told Swiss news website 20min.ch.
Among those killed were 17 people from Switzerland and a three-person family from Austria, police said.
The Junker JU52 HB-HOT aircraft, built in Germany in 1939 and now a collectors item, belongs to JU-Air, a company with links to the Swiss air force, the ATS news agency reported.
"We can assume that the aircraft hit the ground near-vertically and at relatively high speed", Daniel Knecht of the Swiss Transportation Safety Investigation Board said at a news conference in Flims.
The crash happened hours after a family of four were killed when their small plane went down further west in the Swiss Alps.
Officials can essentially rule out a collision with another aircraft or hitting an obstacle such as a wire, and there's no indication of foul play or that the aircraft lost parts or broke up before the crash, he added.
According to the SwissInfo website, citing local police, the plane went down near the Piz Segnas mountain on the afternoon of August 4 at an altitude of 8,330 feet.
Swiss reports said the passengers were returning from Locarno, a holiday spot in southern Switzerland on Lake Maggiore, where they had arrived early Friday.
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An investigation has been launched into the cause of the accident. The heat makes air thinner, increasing the burden on the aircraft.
Almost 5,000 Ju-52 planes were manufactured between 1932 and 1952.
The company's flight operations were suspended, it said.
Chief executive and co-founder Kurt Waldmeier said the planes "are flown exclusively by very experienced professional pilots, and strictly checked and maintained by our own technicians".
"We can not yet explain what led to the tragic accident on Piz Segnas", he said.
The plane was operated by Ju-Air, a Swiss company that offers tours with vintage former Swiss military aircraft.
It said it was suspending flights until further notice.