The r8168 is the panacea of the RealTek NIC drivers that works enough… until it does not.
It could be related to which application is managing your network connectivity. I find that Network Manager tends to break depending on if I am in Debian SID or Arch. Sometimes it will work in one but not the other until an update.
On Arch, I have been using dhcpd more. In Debian, I have the best of luck with Network Manager.
Look to see what is happening on the next reboot by running sudo journalctl -b 1