My first guess is that your hosting company has instituted something like mod_throttle which blocks a connection when more than x simultaneous connections are attempted and that number seems low.
The way downloaders work is that they open 6-20 connections at the same time and fetch things in parallel. This typically puts some strain on the webserver depending on what is being requested. As a method of combating this, many hosting companies use a firewall or apache module that blocks the offending IP for a period of time.
Firefox can be configured to use
pipelining which is set to 8 or 12 simultaneous connections at the same time. By default, pipelining is off, but, many of the 'faster web' plugins will enable it.
You might check with your hosting company to see if they have recently added connection throttling like that. It is also possible that his ISP instituted similar caps, but, that would be much less likely.