Muddiest Point for 10/15-10/16

I have a question about LAN and WAN.  Is it possible to be connected to a LAN and a WAN at the same time?  I was talking over skype with a friend in England when she lost her internet connection and we couldn't talk to one another.  However, she later told me that she still had local access according to her computer.  So I was wondering if you could have both LAN and WAN at the same time, and lose your WAN connection.  I thought you could only have one type of access or the other, not both.