If it's not one thing, it's always another. While I was still uncertain whether or not the modem issue was fixed by getting a replacement modem, my laptop network card disconnected me. I was pretty sure it wasn't the modem... so I disabled the card using XP's tools and re-enabled it. Sure enough, the connection was still there and very strong.
An hour later it happened again.
So I went to Google and tried some keywords, and LO! I found an answer. Apparently one of the XP "hotfixes" has a fairly negative impact on some wireless cards. If I understood what I was reading correctly, and I may well have read it wrong, my choices as of a month ago were to uninstall the hotfix or live without the internet. Fortunately, Netgear had released an update to the drivers on my card and that was supposed to fix the problem.
So I installed the update, not without considerable trepidation, and it works again... and hasn't died at all since the upgrade several hours ago.
One at a time I kill off these problems, one at a time. Eventually I'll have a properly working system again.