You refer to two local area connections. Do you mean you have two network hardware devices in one PC such as one being wired and one wireless? Is this a laptop or desktop? It would be possible to have internet access through a wireless connection if a wired connection was unplugged or visa versa. PCs don't always like to have two local connections to the same network and it's possible one would be automatically disabled though another might work. If you have a wired and wireless connection to a router, only one of the two is going to connect through the gateway. Maybe you're having a problem you are trying to resolve. Post that along with more about your hardware.