Quick answer: "Failed to authenticate" = session validation failed. Use premium account, online-mode=true in server.properties, server must reach Mojang session servers (don't block outbound).
If you landed here: "Failed to authenticate your connection", "Invalid session" or premium players can't join.
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Frequently asked questions (FAQ)
Failed to authenticate your connection – cause?
Usually but no premium account, Mojang session servers unreachable or server can't validate session (outbound firewall).
Cracked server – need>Only for non-licensed clients – not recommended for public premium servers. For cracked: + auth plugin (e.g. nLogin).
Only one player affected?
Client-side: re-login, restart launcher, disable VPN. Server-side: session servers reachable? Check outbound DNS/firewall.