My PowerBook G4 says that it's connected to wifi, but it still no internet connection. The connection will work for a while, and then I'll have to run diagnostics and it will work again for a few minutes and then stop working again. What's going on??

Basic troubleshooting techniques for any problem

1. Restart the computer = joy?
2. Reset the router = joy?
3. Restart the computer in Safe Mode = joy?
4. ask yourself "What changed immediately preceding the onset of the problem?" = Anything... hardware, software, cabling, environment (interference source)

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You could go to system preferences network/airport/advanced/and make sure you have (remember networks this computer has joined) is checked. Then in advanced TCP/IP click renew DHCP lease.