It may be the absence of those number but my sister's recent experience gave me new light on this subject.
She has an iMacDV 400 which is used by her and 3 teenagers so it goes through a lot of abuse. A few weeks ago she called in a panic saying that she kept getting a message that she couldn't connect. Of course she blamed the modem. She brought the computer down to me and I gave it the complete treatment which it needed, new battery, initialized, installed Jag, etc. and reset all the internet prefs. I logged on from my house, my folks and even long distance to her ISP. All OK, right?
Nope.
She got it home, hooked it up and again no connection. I asked exactly what the message said and it was either "no dial tone" or "unable to connect, yadda...". Since I knew it wasn't the modem I suggested she connect to another phone jack. But she insisted that the phone was fine since it is the one she uses for everyday calls. Nonetheless she dragged the iMac to her bedroom and lo and behold it connected just fine on a different line. It seems that the jack or the line itself just got to the point it couldn't handle the modem connection but works fine with a phone. It was odd because she had been able to use that line for years, disabling the call waiting with a special code, then suddenly it just wouldn't work.
So, long story short (yeah, I know, too late for that), try out a different jack, phone line, wire, etc. before you blame the modem.