|Keep It Simple, Stupid|
I haven't explained clearly in my previous post.
SSH is several things. Besides a way to run a shell in a remote machine, it is also a transport protocol that can run several communication channels in parallel between the client and the server over a single SSH connection.
When you use SFTP, first a SSH connection is established to the server and then one channel running inside that connection is open and attached to a SFTP server.
In your particular setup, probably the remote SSH server is configured to accept only requests for SFTP channels.
So, now, the remote server can be limiting the number of incoming connections in two ways: (1) limiting the number of SFTP channels per user, or (2) limiting the number of SSH connections per user.
If it happens to be (2), then you can open a SSH connection to the server, and then run several SFTP sessions in parallel over that single SSH connection.
The interesting thing is that the OpenSSH ssh client, has a mode of operation that makes pretty easy to work in that way, establishing a connection and then running sessions (including SFTP sessions) from other programs on top of it.
The following script would tell you if you can actually run several SFTP sessions in parallel: