As any responsible sys admin, I only allow local connections to my production database server. This means that if I need to access my DB from my local machine I would most likely need to use an SSH tunnel in order to connect.
For some reason, pgadmin3 no longer seems happy on Ubuntu 16.04 LTS and I am not able to make an SSH tunnel. In addition, it is a bit annoying that you are not able to save passwords with an SSH tunnel by default in pgadmin3, especially since my password is a long and random 50 character string.
The solution is pretty simple using the -L SSH flag.
ssh -L 15432:localhost:5432 $USER@$SERVER
This command creates a tunnel on my production server, and forwards port 5432 (the default PostgreSQL port) to my local port 15432. This allows me to connect using pgadmin3 as if the database were running on my local machine.