I had this issue after switching to Verizon and buying 2 Galaxy S 3's. Mine would not allow wi-fi to turn on at al. After my girlfriend was able to connect immediately and after countless hours, I did the following:
1. Downloaded wondershare mobilego for android and backed up my contacts, pics, and apps
2. Performed a factory reset
I was able to restore all of my phone settings and immediately connected to my home wireless.
Hope this helps