If you too have encountered the same error prompt when you tried to connect to your wireless network at home, be sure to check your router settings first and ensure all other devices at home can connect to the network. Doing so will help you isolate the problem, which in this case is apparently within the GS 3 handset.
Here are your options to resolve this problem:
A. Wipe Cache Partition
One effective solutions to this issue is through wiping the cache partition on your Galaxy S3. Here's how to get this done:
1. Turn off your phone.
2. Press and hold the Volume Up, Home and Power keys simultaneously.
3. Wait until the phone vibrates, and then let go of the keys.
4. Press the Volume Down key, to scroll to and highlight wipe cache partition option.
5. Press the Power key to select and wipe the cache partition.
6. Wait until the wiping is completed and go back up until reboot system now is highlighted.
7. Press the Power key to restart the phone.
If problem persists after wiping cache partition, we suggest you try to forget the network, reboot the phone, modem and the router.
If all else failed, the last resort you can try on is a factory reset. Doing so will clear old files that are still left from the old firmware. However, you have to note that a factory reset will erase or remove all your personal information stored on the phone, including your customized settings. That being said, you should create a backup of all your important files first.
Hope any of the above suggestions help you fixed your Samsung Galaxy S3 home Wi-Fi issue.