If I use JSch from more than one thread, how should I use it

As any other non-thread safe class.

Access it from a single thread at a time only.

Use synchronized statement:
https://docs.oracle.com/javase/tutorial/essential/concurrency/locksync.html

If this downgrades performance, you can create a connection pool.

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Though I do not think this exception is caused by a concurrent access.

It's rather that the server rejects too frequent connection attempts from the same host (what is quite common).

Just to complement @Martin Prikryl answer that actually solved my problem.

In my case it was the server that didn't allow more than 20 simultaneous connections. After spending 4 hours, I finally talked to the infrastructure team and they increased the maximum SSH connections on the Linux server to 50.

For the record, the steps are (you'll need root access on the server):

1. Edit the /etc/ssh/sshd_config file and set the MaxSessions parameter to 50 connections (in my case).

2. In the same file set the MaxStartups parameter to 50/30/50.

3. Restart the SSH service (or restart the whole box).