Because I'm from Serbia, and the site is in Serbian, so the five files are named Š Đ Č Ć Ž (Unicode). And my keyboard is made for Adriatic region. Not a pain actually, as I said above it worked fine with my previous hosting provider, but when I migrated the impossible happened, although the new one also uses Linux.
The files are fully named literally as Š.php, Đ.php, Č.php, Ć.php and Ž.php and they all use the "switch case" command, and also they worked well until now when I migrated the site. So, the problem is either in the server my site is hosted on, or in my files's names, that's why I asked here for help to resolve it. The Apache's default charset is set on UTF-8, as my hosting provider just told me, and I added the following at the top of my .htaccess file:
IndexIgnore *
IndexOptions +Charset=UTF-8
DefaultLanguage en-US
SetEnv TZ Europe/Belgrade
SetEnv SERVER_ADMIN dejan@fmdh.rs
Options All -Indexes +MultiViews +FollowSymLinks
AddType 'text/html; charset=UTF-8' html