Let's say I'm trying to monitor a box with Nagios, but there's a runaway process that's stealing nearly 100% of the computer's CPU and RAM. How does Nagios continue to alert me about the status of the box, in this case? What's to stop the runaway process from blocking Nagios from running as well?
SnapOverflow
If the monitored server is overloaded, Nagios or any other monitoring program is unable to work at that server.
But no worries, you will receive your precious alerts: you will recognize the situation about the fact that every single Nagios test monitoring the server will timeout.
Though if you meant you have everything, including Nagios, running on the same server, then your Nagios is vulnerable for sudden death. Always have a separate monitoring server or monitoring will be useless: it will go down on the very moment you actually would need it!
You might also consider putting up some informative monitors, such as load average. Make Nagios warn you if load average gets past some critical point.
Graphing various server aspect such as CPU, memory and disk usage with MRTG, Cacti or Nagiosgraph is also helpful and makes it easier for you to keep an eye on trends.