I was having the same issue as the op on my Samsung Note 2 and was pretty confused by it since my laptop and other peripherals were connected to the same home Wi-Fi. I tried restarting both my phone and router but still the same thing. Forgetting my Wi-Fi didn't work. Taking out the battery and waiting it out didn't work. Resetting app preferences didn't work. As far asI know I didn't uninstall anything between the time it was working (during the morning), to the time it stopped connecting (this afternoon). So I started looking at my Wi-Fi menu options more closely and found a solution that doesn't require a hard reset.
If any of you have a router with a WPS button on it this could work for you too. With your phone on and the Wi-Fi in your list select from the menu options "WPS push button" It'll show a pop up with a bar filling up in blue and some instructions. Now push the WPS button once on the router. You'll see on the phone behind the pop up that it is connecting and the pop up will go away when it connects.
To test if it would stick I turned the Wi-Fi off on my phone and turned it back on again and it connected like normal. I hope this helps those concerned with their router.
Thanks so much, saved myself a factory reset, it totally worked for me!...I was having an issue on my galaxy s5. switch off my mobile data, turn on wifi. Had no wifi symbol display in the top bar of my screen, wifi was acknowledged as being on in my "Wireless Settings" and it showed my home network as 'saved and secured'. I tried and tried to connect, failing that I deleted the network and added it, re-entering my password. It would again, connect to the network, but drop moments later and still no symbol. I then landed on this feed and read about restoring to factory settings (not ideal because its a pain backing everything up!). Then watched a youtube video where someone changes your IP from 'DCHP' to a 'Static' IP which is easily done at the time of actually connecting to a network and entering the WEP key....tried and failed...but before I reset back to factory settings I simply took the battery out of my phone and switched off my router...waited about 5mins..switched the router back on, made sure it had a stable internet connection, then repowered my phone. Connected straight away and was back online. Now Im not saying that will work for everyone and I was F$#%ING shocked it worked for me...but maybe try the simple things first before restoring back to factory settings.
Hope this is a help to someone else lol. ;-)