Why isn't my MacBook connected to the internet?

I've got a MacBook running 10.5.8. I have a constant problem with my wireless internet connection being interrupted, despite Airport showing I have full connectivity. While browsing the internet, Airport will show full connectivity, but I won't be able to surf or connect to anything. Turning Airport off, then back on usually solves this problem. Sometimes I have to do this repeatedly, or it takes a few minutes. Its getting to be a huge pain in the ***/liability when playing games online. Anyone know why this happens, or where to look? Is this my ISP's fault, my router's fault, or my laptop's fault? And how do I fix the problem?