I am actually in the process right now of downloading a test site from a client's server to move it into production. All of the sudden after 1,000 files the server booted me out, saying there was too many connections from my IP address.
One reason I love FileZilla is because you can queue multiple connections. However, I didn't know how to turn it off so I could get to work. Here's the solution: go into your connections manager and select the host you are connecting to.
Check the box "Limit number of simultaneous connections" and drop that baby down to one. It's going to suck waiting the extra hour..or more, but at least work can be done.
FileZilla Transfer settings