dialup test

Telco problems are probably the hardest to get corrected.  Many Telco's don't want to hear that you are having modem problems on the "voice" line.  It will take persistence and lots of calls to the Telco to get it resolved - if it really is a Telco line problem.

However, there are several areas that are the "customer's" responsibility.  e.g. "inside wiring" (wiring inside the residence and past the Telco "demarc" (interface) box, unless you specifically have paid for inside wiring service).  And you need to check your (customer's) side before going back after the phone company.  I worked with data lines for many years and speak from experience.

Depending on what happened to cause the problems to start with, e.g. lightning, it could have also caused damage to "inside wiring", the modem and the telco interconnect cable between the modem and the telco wall jack.  A voice telephone may also be causing problems.

The first thing to do is disconnect ALL voice telephones and answering machines from the telephone line and then try the dial up to see if you get a better connect speed.  If you do, then one of the voice phones is probably causing the problems.  Cordless phones are especially bad about causing slow dialup speed problems. 

Next you can try a new 4 wire cable between the modem and the telco wall jack. 

Finally, connect the PC to a different telco wall jack in the residence and see if that makes a difference.

If none of that helps, then it's back to the phone company.  Keep calling, eventually they will either get tired of you calling or you'll luck out and get someone that is willing to really test the line.  Keep on them about it worked until they "fixed" the line.