Yesterday I uploaded a Rails 5 application with multiple databases to Heroku. I have a hobby-dev postgres add-on. This morning I successfully imported my 7 Postgres databases using pg_dump backups created according to the Heroku documentation.
PGPASSWORD=mypassword pg_dump -Fc --no-acl --no-owner -h localhost -U myuser mydb > mydb.dump
heroku pg:backups:restore 'https://s3.amazonaws.com/me/items/3H0q/mydb.dump' HEROKU_POSTGRESQL_COLOR_URL
Several times I have imported these databases because after a few minutes they disappear. I ran my Heroku app and the first database was successfully accessed but by the time I tried to access the second one I received a 'relation "xxxx" does not exist'. When I went back to my datastore the databases were gone. When I tried to run my app a second time I got a 'relation "xxxx" does not exist' on the first table that I successfully accessed the previous time I ran my app.
I'm not seeing any errors when I look at the data-store for the database. They just disappear. I checked to see if there was a limit to the number of databases I could have with the hobby-dev but did not see any. The row count is under 10,000. Each time I have imported my pg_dump files I get a warning email about the number of rows.
UPDATE 2/17/2017 10:42 AM central: The only thing I have found so far are some posts stating that the Heroku filesystem is ephemeral, and does not persist between dyno restarts. If this is my problem:
I assume you use Heroku PostgreSQL offering (instead of trying to set it up on your own). If that's the case the ephemeral nature of dyno file systems shouldn't be your concern.
I recommend that you first create the seven (empty) databases and see if they disappear or not. You can create a single database with
heroku addons:create heroku-postgresql:hobby-dev
After each call run heroku pg:wait to wait until the database has been provisioned. If the databases don't disappear try restoring your backups then.