I found a fix if you are having trouble establishing a wi-fi connection between your smartphone and a Linksys WRT54G Router. My router is 8 years old. My SIII could not establish a wi-fi connection with it. I upgraded the firmware and performed a hard reset which did not fix it. Here's what I did:
1. Get into your router - http://192.168.1.1. If you didn't already establish a password, leave the username blank and enter "admin" as the password.
2. You should enter Basic Setup.
3. Click on the "Wireless" tab.
4. Go to "Wireless Security. The screen should show the Security mode - (WEP) and the SSID - your( network name) and the passphrase/password There is a button called "Generate." Pressing it generates random 10 digit "Keys" Those keys should have been generated when the router was initially setup.
5. Write down the 4 Keys.
6. Go back to your phone and bring up the network name you are trying to connect with. (Edit)
7. Enter the SSID(Network name)
8. Enter the security - WEP in my case
9. When you get to the "password" field, instead of entering the passphrase or password, ENTER ONE OF THE 10-DIGIT RANDOMLY GENERATED KEYS.
No guarantees but this is what worked for me and now I can work on my phone here at home without racking up 4G data charges. I was fortunate to have found this. Like I said, the router was 8 years old but I didn't want to have to buy a new one. And the Samsung support people did not have any answers for me. They told me to go buy a new router. Give it a try if you're having problems and contact me if I can help.