Able to connect to wireless router but no internet access warning

Does the little yellow signal say "limited or no connectivity" ?
Usually that means that you too far from the source. How many signal strength bars does it show (like 2 out of 5)

I hate to ask the question, but is the unsecured network yours? I have seen people set up unsecured networks for gaming and such that only offer local connections and do not have a default gateway or DNS set up for access to the internet. If you are jumping on a neighbor's wifi because it is open, they might have set up one like this.

Not accusing, just asking the question. If it is your network and you can verify the gateway and DNS settings on it, then I would agree with Steimy and say check the signal strength or try resetting the wireless router and see what happens.