Hi, I work for an ISP & have seen what you are describing with many of our customers. Most of the suggestions previous have been good ones & right down the line of things to check.
From my experience it is due to the modem not making a good conenction with the modem at your ISP. It could be that your ISP has a bad modem or modem bank. If it is a software modem & depending on all the stats of your pc, it could be due to the modem (Software Modem - Uses the CPU & software to do it's job -http://www.tbartlett.org/software_modems.htm ). You can find Initialization Strings for "Soft Modems" on that page also. But, you say the modem will work fine in another pc.
Are both pc's running ME ? The TCP/IP stack should be removed & re-installed as well as the dial up adapter. When re-installing, make sure to add "Client for Microsoft Networking" and set it as the default Network Protocol. If after re-installing "Dial Up adapter", "TCP/IP", "Client for MS Networking" you still have the same problem, I would have to say that it must be a bad modem (modem bank) on your ISP's end or you need to re-install your Windows after a format.
HTH,
Best Regards,
SEARogue