"An expert has told me that modems can't go wrong ..."
You need to find another "expert."
I don't claim to be an expert, but I consider the dial-up modem to be perhaps the part of a desktop or laptop computer most prone to being fried by a small electrical surge (over the phone line). If you have a physically removable modem you could remove and then reinstall it to see if that makes any difference.
I can't guarantee that your modem is fried and that there is no software problem, but that is my suspicion.