Stealing network by assigning MAC address?

Hi folks, my question is is it possible for someone to steal your network by assigning your MAC address to his LAN card? I recently started to recieve a message about a conflict with Ip addresses and my ISP claims that someone is stealing internet connection by using my MAC address. My question is is it really possible? They also saied that if I swith to other ISP, the problem will still persist. And how can I change the MAc address?