Hello,
I can try to help you out (majoring in IT and studying networking right now actually).
You are on the right track. Your laptop, somehow is getting assigned the same iP address as another device. What other devices are in your house that are connected to the router right now? Does anyone in the house (including you) have a phone/computer/tablet/toaster/anything connecting to the internet? If so, it is very likely that other device is trying to use the same IP address as your macbook air.
Here is a good tip to troubleshoot. Turn off everything in your house one by one that's using the internet. Sign on with your Macbook first. You should see that, at that time, nothing else is interfering with your connection. Now, one by one, start turing on other devices. For example, you got an iPad, turn that on and connect it to the internet/router. If it doesn't kick your Macbook air, shut it off and try another device. This might help identify a troublesome device.
On your macbook, did you ever set it to have a static IP address? Did any "professionals" (like Geek Squad) set up your home network? If so, they may have configured your wireless network to have everything assigned with static IP.
If none of your devices have a static IP address, it is a router issue then. You need to set the DHCP on your router to allow multiple connections on a wide ranage of IP's and have the router automatically assign IP addressses to your network devices.