I'm going to approach this from the assumption that it is extremely unlikely that you would get five bad modems in a row. This would only happen if there was some cause such as a conflict in hardware design such as power levels that caused failure or fluctuations (do you have a ups?) etc.
I'm going to suggest that what you may have is a problem with your telephone circuit that is probably in your wall jack or cord from jack to modem.
Your wall jack, if it is of most common designs, is not airtight and every time there is an air temperature and/or pressure change in the wall, (outside wind, doors opening, day/night etc.) is sucking air in and out through the port. Think of it as a strainer for dust and moisture. Your computer has a fan ventilating inside which causes a constant pressure change and if the port of the modem is not relatively airtight, it too will act as a strainer catching dust anytime the computer is on.
Your computer is probably in your bedroom where you sleep, or may be near a bathroom or kitchen. The dust by itself won't cause much trouble, but minor increases in the moisture level that you probably wouldn't notice (from breathing, cooking, showers) are enough to cause corrosion; the dust acts as a bridge for the moisture, contributing to corrosion and run-on sentences, which may be causing the resistance between the metal-to-metal contacts of the jack and cord to increase, (dirty open, which is what I think is happening), or the corrosion may be causing a high resistance short, usually accompanied by static, which can be at low levels and not be obvious (or may be what you are hearing) or both. Also, the baseboard type jack is prone to this problem, especially when it turned upwards so that foreign material falls into it.
So, what to do? Remove the cord from the wall and inspect it. The contacts should be shiny gold and clean. if you see a dark spot in the plastic under the contacts , moisture is a definite problem and the cord is garbage. Any blue-green appearance is a sign of corrosion.
Look also at the modem end although the trouble is less likely to be there. Now its just me, but I'd replace the cord just to eliminate it as a cause of your poblems. I've seen them cause many problems from being stepped on, chairs rolled over them, melted by heaters etc. And they're cheap. Don't waste your money on high falutin gold plated super duper cords, just don't get one longer than you need and don't coil up any extra.
Now look at the jack. Again, only clean, shiny straight. wires that look like they are not bent, and under enough tension to make a good contact when the conector is in place. Sometimes, a wire is weakened and doesn't make a good connection. Or has been bent or out of position. If it doesn't look clean and straight, you may have a problem there. Same with the modem ports.
I think that when you experience a problem, your "test " of plugging the phone into the wall may be causing enough movement that the corrosion is temporarily rubbed enough to allow a good enough connection for the phone to seem to be working fine.
The next time it happens, wiggle the cords, unplug and replug, and see if starts working again. Then we can get into the exotic stuff.