Background: The dmesg buffer contains only kernel messages. Not messages from user processes. Those go eventually into /var/log/messages. In reality, that's all MUCH more complicated, and has to do with the syslog configuration, see man syslog, and read /etc/syslog.conf and the files in /etc/syslog.d.
During booting, when the init system comes up, one of the tasks it does is to save the dmesg buffer (the kernel messages) from booting in that /var/run/dmesg.boot file. You can see exactly where that happens, the script is in /etc/rc.d/dmesg.
Typically, networking errors happen during the initial startup phase of the system, when the other rc scripts are being executed. Typically, those error messages are from user processes, not from the kernel, so they won't be visible in dmesg to begin with. And I don't remember the order of rc files by heart; the network startup might happen after dmesg.boot is written.
I agree with PMc's advice: Read /var/log/messages, and configure syslog.conf to log whatever you need. And attach a physical console, and see the messages stream by.