Concatenating an already existing filename to create a new file
Prefer RAII-style opening the file in the constructor. You already done that with

    std::fstream fileIn( originalFilename, std::ios::in );


so why not follow the suit with

    std::fstream nextFile(nextFilename, std::ios::out);

There is no need to explicitly nextFile.close();. The destructor will take care of it at the end of each iteration.
There is nothing wrong with the early return. Instead of setting originalFileFound = true;, and testing it later on, quit immediately if opening fails. That would spare a boolean flag and a level of indentation.
Consolidate handling of nextFilename in one place. Consider

    if (!fileIn.is_open()) {
        print_error_message();
        return 1;
    }

    std::string prefix = originalFilename.substr(0, originalFilename.size() - extension.size());

    for (int i = 0, counter = 0; counter &lt; 5; i++) {
        nextFilename = prefix + "_" + std::to_string(i) + extension;
        ....
    }

I would be very cautious to use this program, because it may overwrite existing files.