That's probably by Article 9 paragraph 2 of the convention: It shall be a matter for legislation in the countries of the Union to permit the reproduction of suc

That's probably by Article 9 paragraph 2 of the convention: It shall be a matter for legislation in the countries of the Union to permit the reproduction of such works in certain special cases, provided that such reproduction does not conflict with a normal exploitation of the work and does not unreasonably prejudice the legitimate interests of the author. The European Union Copyright Directive explicitly allows member states to have legislation allowing copies for private uses. Germany for example makes use of that exception. It is allowed to make copies of copyrighted works for personal use (with some restrictions). In exchange we pay a special tax on storage media which goes to right holders (for example 36€ on a smartphone). That topic is far away from my area of expertise but we had some high-profile court cases regarding private copies and if there had been any way to shut that down via the Berne Convention it would surely have been tried.