Bogus NUnit Error: 'Could not load file or assembly nunit.core'

After adding a set of unit tests to an existing test project, NUnit started throwing this exception on our build server that, of course, wasn't being thrown locally.  I was perplexed since the balance of the changes were a new test fixture, a couple of new references to the test project, and an app.config change.  How could any of those things cause NUnit not to find... itself?

I knew the error message was probably not indicative of the real problem, and my suspicion was confirmed by the first search result returned by Google.  Basically the problem boils down to an error in the app.config file.  In my case it was because the config section I had added was defined in my (customized) machine.config file, but not in the copy that was on the build server.