Edit: Originally this post mentioned the old auth servers had been taken offline. Apparently, that issue wasn't related with the alpha/beta auth servers being taken offline, as originally mentioned here, and has been, since then, fixed. This post isn't applicable anymore.
The original post is kept below for historic reasons.
You might have received a "Failed to login: Bad login" error when trying to connect to a server on some older modpacks. This is because Mojang has disabled their old authentication servers (login.minecraft.net), which means that Minecraft versions older than 1.6 can't authenticate anymore. For example: Tekkit Classic.
If you're running a Minecraft server for these versions, the only workaround right now is to set your server to offline mode (
online-mode=false) via the server.properties file located in the root of the server folder, which will disable all authentication, so it's highly recommended to install a plugin like AuthMe (https://dev.bukkit.org/projects/authme-reloaded) so your players are at least acceptably safe.
This issue is listed on Mojang's bug tracker at https://bugs.mojang.com/browse/WEB-1429.