Cable Modem or ISP Problem?

I have a relative in Louisiana using Cox for her ISP. Her service has been spotty this weekend, although she did nothing to start this sequence of events. She called Cox and was told that her cable modem had died and that she should buy another one. This struck her as odd for two reasons: first, the was renting the cable modem from Cox; and second, the cable modem did work, only sporadically. Currently, if she stays in front of the computer, she can maintain her internet connection. If she walks away, however, the connection is broken and she has to unplug both the power cable AND the coaxial cable from the modem in order to establish another connection, which she must then not walk away from.

She is now in the situation where, after countless calls, Cox literally won't talk to her, which is why she asked me to start this thread. Here is her question: is this a cable modem problem or a problem with the ISP?