The backups are 250MB. I don't think that's very big, but it appears the problem is increasing with the size.
Log from the Backup gem below.
Note the time span; about 37 min into the uploading I get connection reset.
[2015/10/30 09:20:40][message] Storage::S3 started transferring '2015.10.30.09.20.01.myapp_postgres.tar' to bucket 'myapp-backups'.
[2015/10/30 09:57:06][error] ModelError: Backup for Back up PostgreSQL (myapp_postgres) Failed!
[2015/10/30 09:57:06][error] An Error occured which has caused this Backup to abort before completion.
[2015/10/30 09:57:06][error] Reason: Excon::Errors::SocketError
[2015/10/30 09:57:06][error] Connection reset by peer