Have you checked that the modem or network card is not going to sleep or automatically shutting off?
right click My Computer-->Properties-->Device manager-->
-->Open Network adapters-->right click your network card-->properties-->
Check under the advanced menu and the Power Management menu
Make sure that your network card isn't going to sleep automatically.
Alternatively, you said you were on dial-up, maybe check the properties for your modem
driver also in Device manager.
Do you have any yellow exclamation marks in Device Manager?