Internet doesn't work with apps without using vpn
I just had this issue on my Nexus 10 running Android 5.1.1.

I could not connect to the internet via my Wifi access point (AP) but I could connect to my VPN (Private Internet Access) and then connect to the internet. This only affected one device, all my other devices (laptops, phones, other wifi gadgets) had no issues connecting to the same AP.

The issue I found had to do with the DNS Leak Protection from Private Internet Access (PIA). The solution was to set a static IP address and DNS settings.


Disconnect form your VPN
Long press on your wifi network in the wifi settings
Check Advanced Options
Add your IP address and gateway (the IP address of your router)
I added 209.222.18.222 as DNS 1 and 209.222.18.218 as DNS 2. These are the PIA DNS settings.
Save those settings and test your connection


This fixed the issue on my device. Curiously this issue only affected my Wifi AP, when I connected to another access point I did not set have to set a static route.