Postgres 9.6 on AWS with frequent "could not connect to server: Operation timed out"

I've just set up a new Postgres 9.6 on a db.t2.small with 100gb SSD in AWS Sydney region. I can connect to it through psql and pgAdmin using the master user and password, so it has been setup correctly, and the security groups allow access. But after running a few commands it throws:

psql: could not connect to server: Operation timed out Is the server running on host "xyz.ap-southeast-2.rds.amazonaws.com" (54.xxx.xxx.xxx) and accepting TCP/IP connections on port 5432?

So given that I can connect and can issue commands like CREATE EXTENSION postgis successfully, why would it drop the connection so frequently?

I am trying to restore a 300mb dump file, but it drops connection even when running basic SELECTs, or refreshing the object list in pgAdmin. When attempting a restore, it drops connection in less than 30 seconds.