Have "bad login error" when logging into servers

Whenever I try to log into a large public server (such as Hypixel's server, for example), I get a message that says "failed to login: bad login". I believe this may be related somehow to having connected my Mojang account to Twitch for streaming with the new snapshots.
I changed my password in an attempt to fix the problem, to no avail. This has been happening since yesterday.
Steps to reproduce:
1. Open Minecraft
2. Try to login to a large public server (seems to be almost every server I try logging in to, most of them have over 100 slots)