More specifically, you cannot add new abstract (aka "pure virtual") methods to an existing class/interface/trait when that class/interface/trait is already extended/implemented by code you don't control.

Rust lets you combine multiple traits per type.

C++ lets you inherit from multiple classes as well. I don't see how this has anything to do with being able to add new methods to existing types.

There is an important difference: in Rust you can write a new trait, with new methods, and impl that trait for a type without having to change the existing structs/enums which comprise that type at all. You can also do this (with some restrictions, "the orphan rule") for types defined in another library.

In C++ classes, you have to be able to modify a class definition to extend it with new methods (or a new ancestor to multiply inherit from).

You can add traits (with their methods) to existing types, without modification.

C++ classes are types. Rust traits are applied to types.

So hypothetically, if you could add new methods to an existing class, it would solve the problem?

New virtual methods, yes.