An update. I found a couple of programs that let you keep tabs on your open internet connections. NETSTAT.exe comes with windows. TCPVIEW is a GUI version - a free download.
So I get on the 'net, take a look at TCPVIEW and see a bunch of connections. Start up my BT client, start some downloads and see a WHOLE bunch of connections <g>. I don't see where any of them can be "blamed" on Avast.
I let the BT client grab as many connections as possible and, like my orginal post said, I run out pretty quickly. I reboot (so I can test with a clear start) and as soon as I get to the desktop I wipeout Avast from memory (using a program called APPSWAT). I have to do it this way because there's no obvious way of quitting Avast completely short of uninstalling and rebooting. I then log on the internet, fire up my BT client and start downloading. Many more connections are now available. Over 70, just like before.
So it appears that Avast is doing something that LIMITS my maximum number of connections without actually using them. (remember I have the automatic updating turned completly off)
I've noticed something else that may be connected. My windows swapfile is usually around 196 meg. From time to time it'll get real big (over 300 meg) but after a while it moves back down to 196. Since I've installed Avast it seems the swap files gets real big more often. That may be just a coincidence or a false perception however.