modem connections can be flakey, and good connections at a premium. There is hope for better connection stability soon, but in the meantime, try cleaning out your partial downloads and limit your downloads/searches to one or two at a time. Making your searches as specific as possible should also help.

If hosts are dropping your connections, you can encourage them to keep the connections alive by sharing files they want. That way others will also work to keep the connections to you alive.