I wasn’t aware of the “network-test” command in Debian, but it’s actually quite neat:

INFO: This system has exactly one default route
INFO: Host localhost answers to ICMP pings
INFO: Loopback interface is working properly
[...]
INFO: The wifi0 interface has tx and rx packets.
INFO: The router 192.168.2.1 is reachable
INFO: This system is configured to use nameserver 192.168.2.1
INFO: Host 192.168.2.1 answers to ICMP pings
INFO: Dns server 192.168.2.1 resolved correctly www.debian.org
INFO: The nameserver configured for this system works properly
INFO: System can reach Internet host www.debian.org
INFO: System can access web server at Internet host www.debian.org

It does a quick network debugging output. I’d s/Dns/DNS/ though.