First, an infrared camera shows a jacketed bullet leaving the muzzle at 320 C (608 F). [Update: unfortunately that link no longer works, dang it!] That's hot enough to melt the lead core.
Then this thread talks about military studies that confirmed the lead core would sometimes get hot enough to melt:
In early testing of the first Cordite loads for the .303 British they found that if there was no residue in the bore at all, as in a cold clean barrel, the bullet jacket already hot from the high temp gases would often lose a good deal of its lead core while passing through the bore.
I suspect that the very hot charges of high temp Double Base powders quickly eroded the throat and bore, so that gas blowby heating quickly became the major factor in jacket failure.
High speed photos I've seen of .220 Swift blowups show the lead core turing into a steak of vaporized lead, practically disappearing within a few feet of travel.
Friction that produces heat that exceeds the melting point of lead. This result is observed most often by the puff of “smoke” that will be within the first hundred yards from the muzzle. The “smoke” is in fact molten lead. The puff of molten lead does not always occur during this failure. A core that becomes even slightly plastic will not make it to the target properly.
We have known for a while that making the jacket thicker does not make the jacket significantly stronger. As it turns out, we were looking at it from the wrong point of view. We had been looking at a thicker jacket as being a tougher jacket and this just isn’t true, however when you have a thicker jacket you are moving the lead away from the source of the heat,
Barrels longer than 28” are capable of produce failure-creating heat. Remember that the bullet is hottest at the muzzle. The more metal the bullet has to travel over, the hotter it gets.
Six groove, cut, squared off rifling produces greater friction than a 5C or 5R type barrel. The 5C or 5R type rifling produces more friction than a three-groove barrel.
-- yes, under some circumstances a bullet can get hot enough to melt lead.
-- one hot spot is where the bullet is engraved by the grooves
-- another hot spot is the base of the bullet
-- a thicker jacket helps insulate the lead from the heat (or in the case of cast bullets, a thicker gas check, or a thicker coating)
-- gas blowing by a poorly sealed bullet may contribute to heating and melting
-- the bullet gets hotter toward the muzzle. That may explain why some (not all) cast loads foul more near the muzzle. Conventional wisdom is that the bullet was "running out of lube" by the time it reached the muzzle, but perhaps it was simply hotter by the time it reached the muzzle?
Bevel base bullet fired from 357 mag Contender at approximately 2000 fps, showing what sure looks like melting on the corner of the base. Not every recovered bullet looked like this, but some did.