Inductance from electrical lines and radio transmitters wreaks havoc with phone lines. And avoid using phone splitters of any kind. Most phone lines can actually handle something in the 26400 to 28800 range, but if you have a 28.8 or faster modem, the two modems may be negotiating at speeds higher than that which can be supported by the phone line. You can set the maximum speed for your modem to something 26400 or lower to prevent this. Not true at all.
Data connections are modem specific.
X, Y, or Z brand modems are poor. Generally, modems are reliable devices - it is the support offered by the manufacturer that can be poor or difficult to get. Any modem you have should be able to connect to NCCW Online, it just may take some fine tuning. The default setting is usually around 14. A generous setting of 100 should significantly reduce connection breaks.