It seems as though we are discussing several different problems here. Some find distance an issue, some find their phone doesn't even "see" the network, some are getting "unable to join messages" when they attempt to connect.
The latter was my issue (along with the occasional "invalid password" message when I know the password was correct) and I seem to have resolved it by resetting my router/modem (pushing the hidden button on the back) which totally deleted the old network. I then created a new network with a new name and so far everything works as normal now on both my phone, and my son's. We are going on day 3 of a normal wi-fi connection and I have my fingers crossed that it will last.
I agree that there is no explanation of why a router change could be a valid fix since every device would seem to have to be affected to indicate a router problem. At my house my computers all connected to the wi-fi fine and it was just a problem with the iPhones after the iOS 7 update.
I know there is some hostility out there towards the idea of having to change settings on the router to fix this and I won't enter into that discussion. I just want to put out there what seems to have worked for me in case someone else wants to try it.
PS. My phone was always able to connect normally to public wi-fi networks. The problem only occurred at home. For those of you who can't connect anywhere, obviously this fix won't help.