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It turns out my available memory was running all the way down to zero. This only happened while the large file copy was in progress. I was using xcopy to move a bunch of files over.
After searching for possible clues as to what would cause the issue I discovered that my Windows "System Cache" was growing to over 2GB. This cache is responsible for caching among other things files from the filesystem.
So I tried disabling anti-virus, write caching, and registry adjustments. That did not help.
I then found a tool that will let you temporarily reset the size of the system cache to a value you specify. You can't force a permanent maximum (except in Windows Server 2003 or later), but you can call it via a scheduled task if desired. It takes affect immediately and relieves the memory issue.
The tool is called CacheSet and can be obtained from Microsoft (SysInternals).
It isn't a perfect solution since you have to keep running it to keep the cache size down, but it is better than nothing at all.