Any way you can check this modem, to see if it does the same thing on another machine? Also, you know, I've had similiar problems that were fixed by simply dialing a different number to my isp. Worth a try.
You can also try this:
Delete your modem from the device manager. Shutdown and restart your machine, it *should* detect it when it comes back on and automatically reinstall the drivers (but I'd be ready with the driver disc in case, or obtain the drivers before you delete it). Sometimes the settings or drivers get messed up, for one reason or another, and this will reinstall your modem to default values. You can also get another modem (LTwinmodems are pretty reliable), they are all about $25 after tax and maybe even cheaper if you look around.
Another thing to look at: In my town, there's a computer shop that sells a lot of used hardware that they test (so it does work), and it's very cheap. You can take a look at getting another modem that way.