Annoyingly enough, its considered an acceptable “standard” for servers to return a false 404 error instead of 403 Access Denied/Forbidden: https://en.wikipedia.org/wiki/HTTP_404#Phony_404_errors
I don’t know why except Because Security? but it sure makes debugging difficult.
Considering that it suddenly starts returning 404 after 5 uses, I’d have to suspect some sort of security block.
I would recommend that you check the following:
1) Can you see the /xmlrpc.php file at the root of your WordPress installation when accessing your site via FTP? If you don’t, then please try re-installing WordPress. You can do this by going to the Dashboard -> Upgrades page, then click the “Reinstall WordPress” button.
2) Do you use any security plugins that may block access to this file? If so, could you try to disable them? You might also want to check your site’s .htaccess file for any rules that might be blocking access to the xmlrpc.php file.
3) Does your hosting provider block access to this file? If you don’t find any plugin that may block access to the file on your site, I would recommend that you get in touch with your host.