This probably only affects a few, but it's rather annoying. Details here:
https://www.raspberrypi.org/forums/viewtopic.php?f=28&t=138631&p=1085534&hilit=qos#p1085534

Short version: Some(?) Raspi 3 stop responding to SSH after the (correct!) password has been entered.
Attaching a keyboard works, so it's just SSH.

Adding the line
IPQoS 0x00
to the file
/etc/ssh/sshd_config
solves the issue.

I have no idea what IPQoS is, but I do know it solved the issue of not being able to log in remotely at once. (After restarting the ssh daemon or the raspi.)

[edit]Note: this concerns a fresh install of OctoPi!