If you are getting the "static" after you hear the "dial" tones, or you get a recording that the call can't be completed,it would suggest there is a telephone line problem. 

However, the telephone line problem coud be external to the residence and the telephone company's problem or it can be internal (in house) which is the customer's responsibility.  Addtionally, a voice phone i the residence, especially cordless phones or answering machines can also cause connection problems.

However I have to think it's a telco line problem and calling your local phone company would be the first place to start. 

The modem would be the least suspect item, and from your description I would guess that the modem is OK. 

Thanks for steering me away from the modem and toward the telephone, Fireberd. You're right.

I still don't know exactly what the problem is, but I accidentally discovered that simply removing the handset from the telephone and laying it on the desk eliminates the static and allows the connection to be made. Of course, unplugging the telephone completely also works.

I'll be the first to admit that this bit about simply laying the telephone handset on the desk and letting the dial tone buzz away while I make my Internet connection sounds more than a little hokey and could be some kind of coincidence that I haven't figured out yet, but so far it's worked about a dozen times when I couldn't make a connection with the handset in place on the telephone.

The telephone and the computer are on separate lines, but are tied together at the box on the outside of the house. There are no other devices connected to our telephone line.

I guess the problem is in the telephone itself? Weird, huh?

Thanks again.