You can have a Wifi connection without having an internet connection.
Try the following:
1. First try resetting the device itself (nothing will be lost): Hold down the Home and Power buttons at the same time and continue to hold them down until the Apple appears (up to 30 seconds). Once the Slide to Unlock screen displays, see if the issue is resolved
2. If that doesn't resolve it, go to Settings>Reset> and click on Reset Network Settings. You will have to put in your Wifi info again, but that may clear it up
I had an issue yesterday with my iPhone 5s. Wouldn't even connect to Wifi even though my iPhone 4, and every other device that uses Wifi were working fine. Tried several times to reconnect with no luck. Finally Reset Network Settings, and as I was setting it back up, reset my router for good measure, and then made sure that when I tried to connect again, that I was sitting right next to my router. Finally got connected. I'm not sure if the issue was my phone or was Time Warner, but since other Wifi devices randomly seem to encounter this issue - my money's on a Time Warner issue.....
Best of luck 🙂
GB