I was going to comment on the same. I don't know if that was ever proven. But after Chuck made it official, and the implications and symptoms were revealed, it is said that some pilots of the F-86 made that claim. It seems plausible but is worth noting. The F-86 was not designed for that speed and I do not recall the claims ever being affirmed by engineers. I'd be curious to know, however.
The need for full-motion control surfaces was discovered around the time — or soon after — the crash of the de Havilland DH 108 "Swallow" that killed Geoffrey de Havilland in England.
The need for full-motion control surfaces was classified top secret, and was one reason films like this one were not released to the public for many years after the event.
If the F-86 was taken supersonic, it would have been very risky to the pilot. I'm not saying it didn't happen; actually, the sound barrier had been closely approached if not broken by pilots of the P-38 Lightning, when in steep dives, during and immediately after WWII.
The F-86E and later models featured an all-flying tail, which is required when flying at supersonic speeds.
Chuck Yeager discovered during the run-up to supersonic flight in the X-1 that the conventional stab/elevator became unresponsive and was unable to control the aircraft in the pitch axis. Long-story short, he used the elevator trim tab, which changed the angle of incidence of the stab in small increments. This allowed him to control the aircraft during the transonic/supersonic phases of flight.
This information was instantly classified as top secret.
When applied to the F-86, this feature allowed it to safely fly beyond the speed of sound. The Soviet MiG-15 did not have an all-flying tail, so it was limited to subsonic speeds.
Given the swept wing with its delayed onset, courtesy of obtained German research, it is plausible, if:
Adequate altitude.
Manageable buffeting.
Controllable surfaces.
As others have noted, the controllability problem gave rise to the all-flying surface.
If the MiG-15 had had that, I suspect it could have gone supersonic rather easily in a steep, high dive, since it had both the swept wing and a T-Tail that should have minimized elevator turbulence.