error 678 the remote computer did not respond, modem problem?

I have been having constant problems with my Inspiron 9100 with it's connection to my dial up internet. I contacted Dell about this a couple weeks ago and they said that it was an issue with my internet service, and to contact them. So I did. I went back and forth with them trying different things with strings and settings and what not until they finally told me to bring my computer in to their headquarters and they would service it to find what the problem was. They had it for a week, going through it and cleaning up my computer for anything that they thought could be the problem. It's a clean computer now. They reinstalled the modem and all of that, and it didn't fix the problem. I got it back yesterday, and you know, it stayed connected for amazingly 4 hours. Then I got the 678 error. Back to where I'd started. I then couldn't get it to reconnect. With the exact same phone line I plugged it into my other computer and it has worked fine and could stay connected for days. So it's not my phone line, so it's something with my computer. What is the problem? I'm not happy with my computer much because not only is it working really slow, but I've had this modem replaced once before already back in June, and I've only had it for about 9 months. Any thoughts?