I just want to share something here, which may help others.
On updating a WP multisite install from 3.7.1 to 3.8 I got this error message for each site I'm hosting: Your server may not be able to connect to sites running on it. Error message: couldn't connect to host
The server is running in virtual machine, and accessible through Apache proxy from another VM. I am using domain mapping.
So apparently the upgrade network script is trying to access the sites I'm hosting, which does not work because the VM is unaware.
All i had to do to fix this was to add a line to /etc/hosts, containing a list of the hosted domains:
127.0.1.1 domain1.com domain2.com domain3.com example.net
The upgrade network script ran smoothly afterwards.