On dial-up, the most likely problem is with your Internet Service Provider. If they've oversold their service, they may have to look for somebody to disconnect when a new user tries to get on. They usually look for the line that's been "quiet" for the longest, but may not recognize activity such as downloads of "large pages."
This doesn't sound like your immediate problem, but if they're overloaded, they may not be able to give you a connection immediately when you dial in. Your computer times out, and you get the "Page Not Available" message.
Sometimes an ISP will have alternate local numbers that will be less heavily loaded, so you might try contacting your ISP to see if there are choices. If not, with dial-up you seldom are locked in to long term contracts, so you might tell them you're considering a change if service doesn't improve.
If you really think that your problem is with the phone line, and not just with the ISP, there are a number of web sites that offer "modem check" utilities. One, as an example only is at 3COM, but is likely to involve a toll call. Description at ModemSite. I have used 3COM's online tests some years ago, but don't know anything about their current setup. My last pass there, several years ago, was to a site that checked only the basic function of the modem. This one is described as testing the line quality as well. (But it may only work with 3COM modems?) ModemSite is just one I found via Google, and I can't really offer any opinion on whether they're reliable. You could likely find similar info, and maybe a local connection, by going directly to 3COM, USRobotics, or your own modem maker's site.
Perhaps, knowing that this kind of service is available, you'll be able to search one up, if needed. USRobotics is one other very popular modem maker, but I found a Google hit indicating that their modem test site is/was currently offline. They promised a "back up as soon as possible," so you might find something by going direct to their site. Your own modem maker may also have something online.