I believe that I've finally figured out the darn modem.
After a bout of close observation, the situation degenerated to the point where the modem was shutting down every morning, and I would usually have to spend the next few hours getting it to work again. These "repairs" would involve any combination of hitting the device with a stick, restarting it, re-plugging it, and rebooting the host computer.
A few days of careful experiments later, however, it turned out that the best recourse was to wait.
I suspect that the darn-fangled device tends to shut down overnight for one reason or another, and therefore needs to be jump-started each morning. When I start it up, however, it apparently takes a while to "prime" itself -- which means that if I boot up the computer immediately afterwards, it won't detect the modem just yet.
I don't know how long it takes before the modem actually gets up and running, but I've found that I can usually start off a game of Zuma or two, then reboot the computer after a couple of game overs. Chances are good that the modem's working by then.
With that said, I believe that the modem is dying. It's not supposed to start up this slowly, I think, and in any case I'm still encountering the periodic disconnections. Technical problems aside, I also have to realize that it's a 56k dial-up modem, which isn't quite the pinnacle of Internet capability nowadays.
So I'm in negotiations for a cable internet connection right now. I'm talking to ICable with regards to their services, and they seem pretty happy about my willingness to pay in advance for a year's worth of fast Internet access. Globelines is also a possibility at this point, but I'm probably going with ICable for their customer service capabilities. That, and Globelines might require that I wait anywhere from two weeks to two months for setup, despite the lower prices. I know that I'm staying away from PLDT at least, if only because they always seem to encounter problems with either connection, or services, or both -- it's like the 1980s all over again, when they took ten years to install new phone lines.
Regardless of who'll be taking care of the faster connection, however, this will eventually result in one thing: I'll eventually have to put the modem out to pasture.
This is not to say that I have any regrets about doing so. In fact, it's the complete opposite: Once I disconnect the stupid device, it's going to take all my willpower not to drive it to the middle of the nearest empty field and smash it with a good-sized sledgehammer. But because I have certain pack-rat tendencies, I'll probably just end up putting it back into its styrofoam box and storing it in my closet.
As with many things, however, it's the thought that counts.