You will get this error if you have Minecraft logged in (other Modpacks are available), and then you try and log into your Minecraft account with Tekkit. It automatically prioritizes the first logged in (Chronologically), and so it will throw the error "Failed To Connect: Bad Login" at you whenever you try and connect using Tekkit.
So, to fix this, close Tekkit and Minecraft (and the launchers automatically popping up after closing!), and open up Tekkit again. Should be fixed.