Hmm, downloading is certainly activity, and your ISP shouldn't be disconnecting you for that (unless it's a budget ISP, then they might), but ME might not see that sort of activity. If the message saying you are going to be disconnected appears in an Internet explorer window, or a java scipt window(and the below dial up procedure show no auto disconnect enabled), then it is almost certainly your ISP.
With dial-up:
Try IE->Tools->Internet options->Connections->*higlight your dial-up connection*->Settings->look for an option involving auto disconnect
With DSL/cable, it could be your ethernet card going to sleep (although activity (even d/ling) would almost certainly keep it awake, and it wouln't give you that message). Go to Control Panel->System->Device Manager-> find ethernet card->*right click*->Properties->Power Management->uncheck "allow coomuter to turn off device"
I just ran through both of those on XP, so the process my differ slightly on ME.