AT&T - SAMSUNG NOTE 3; Rooted/Exposed/Lucky patched.
I'm starting to have the same issue with my home wifi. It's never done it before aside crappy public wifi's
but never my home wifi. I don't think the issue has anything to do with manufacturer brand or service provider.
Mostly because for me the issue didn't start until after messing around with the build.prop and other kernel and system settings.
Cause it's also doing that thing where it makes you think its turning on wifi but just ends up staying off... switching airplane mode on/off didn't seem to help unless done right before a full reboot and I immediately turn wifi on before any appsONboot had a chance to activate and mess with anything. Cause it's also doing that thing where it makes you think its turning on wifi At one point logcat reported corrupted wlan0 driver. It also was saying certain apps were either not responding or sending their intent as user3 for user0. (As if I have my phone set up for multi users and the wrong person account responded to the app)
*As a small attempt to find a real underlying definite problem. I'm going to wipe&flash a shiny new Android OS with ODIN
then step by step recreate all tampering I've done including my goofs like clearing A tuners app data before going into the app and resetting all system changes to original.
Hopefully I can recreate the issue and pin point what exactly is the cause.
Next time someone has that issue they should take and post some logcats so we can find a better understanding and quicker resolve
It can't be the modem/router if it happens on multiple wifi. BUT if it works after a quick modem restart, it could be the ANDROID is having DNS issues with requesting an IP addy when connecting to a hotspot. and is only wanting to use 1 static or few most common IP assigns. IE. xxx.xxx.x.1 (through 5 or so). My reason for that assumption is if it works after restarting the modem and clearing out all previously claimed IPs by other devices, then giving your phone first dibs to quickly claim the previously taken IP. That means it can connect, it just cant comprehend not being first in line. Besides even if the modem is unplugged, that wouldn't stop a device from connecting to the router at all. it would just stop the router from connecting to the outside world.
If you want to change router settings or see what its doing open any folder/web browser and type
ftp://192.168.0.1 Username: admin, Password: admin (Unless otherwise altered)
Disclaimer: I could very well be absolutely wrong and have no idea what I'm talking about on all counts.
You'll have to excuse me, I'm from Oregon and there are 3x more rec pot dispensaries in my town than there are stop lights, so what ya gonna do.

~ Spoot