O M G !
Finally I have connected again!
Thank you for posting this solution - I was burning through my data allowance for the month!
Such a simple and elegant fix!
I have all of the updates and had tried all of the other solutions resetting passwords and network settings, restoring my iPhone (5S) through iTunes and losing some data since my last back up, cycling power on the iPhone and the router, resetting the router, re-entering all the MAC addresses on the Permitted List.
I don't have an Airport device, I have a lame router but that solution still worked for me!
I switched off the security, applied the change, then switched the security on again and applied the change and the iPhone connected immediately after I typed in the (always correct) password!
For those of you who have a regular router - not an airport device - you need to connect to your router (usually 192.168.1.1), enter your password and then navigate to your Wireless LAN configuration. Note your security setting (WEP-WPA-WPA:PSK, etc.) and change it to NONE then APPLY that setting. Now change the security setting back to whatever it was and click APPLY again. Now select your wireless access using your iDevice and enter the password again.
CONNECTED!
Yay! [does a little dance]
<Ahem> - please forgive my exuberance!
BTW, I have LIKED your post in the hope that it becomes part of some ordered list and this solution rises through the ranks. - I hope it does.