Can the 'SSL Handshake Failed' error be caused by a problem with Hostinger's server itself?

Yes, a problem with the Hostinger server can definitely cause this error. This could include server outages, SSL certificate problems on the server side, firewall restrictions preventing cloudflare from connecting, or issues with the server's SSL configuration. If the server is not configured to accept SSL connections correctly, cloudflare won't be able to establish a secure handshake. If the server is down, a handshake cannot be established.