This app was mentioned in 9 comments, with an average of 1.33 upvotes
While it doesn't deal with the cause of your problem, you can use apps like Wifi Fixer to force the connection to stay on until you find what the real culprit is.
https://play.google.com/store/apps/details?id=org.wahtod.wififixer Have you tried this? I have had a similar problem since the update, this fixes it. Sometimes it will still say "no internet connection" but you should be able to use it anyway.
This is a relatively old post, but I found it while searching for something else. If you're still having issues, you can plug your Android device into your PC over USB, and use ADB (Android Debug Bridge, from the Android SDK) to get logs messages and filter for the network related ones. You'll probably get an authentification failed message, which may not be helpful.
Is your problem that it'll work for a while, and then "drop" while still saying it's connected? I used to get that on some (but not all) of my Android devices. Usually disabling WiFi and turning it back on would get it working again for a little while. One workaround is to install WiFi Fixer to automatically reset the connection when it breaks, but it's kind of a pain.
What I found worked was changing the WPA/WPA2 configuration and trying each combination of AES/TKIP (at least I didn't go down to WEP) until finding one that's stable. The router I was using is out of service, hence my searches, but I think there's a problem with Android's AES handling. It may also be in combination with Tomato specifically, since I was also using that firmware, but didn't seem to be having problems elsewhere.
I don't know if you're still trying to figure it out or if this helps, but maybe it can give you a direction to go in.
I hard reset my router. At first I was getting the same issues. I then downloaded an app called Wifi Fixer (https://play.google.com/store/apps/details?id=org.wahtod.wififixer&hl=en) to see what was going on (since none of the native apps give you any information). For now it's working well enough that I could reinstall all my apps.
I seem to have resolved this with the actual reset of my router. I had to setup my SSID and password again. However, I also used an app called Wifi Fixer, which is free on Google Play Store (https://play.google.com/store/apps/details?id=org.wahtod.wififixer&hl=en), and I kept an eye on when the disconnections were occurring. Finally got my phone set up without the SIM Card in.
This app might help: WiFi Fixer
Do not root your Device. It only makes things worse.
https://play.google.com/store/apps/details?id=org.wahtod.wififixer&hl=en .
The above is an apk that claims to fix this problems.