Fortunately, my friend Google never leaves me hanging. The first quick thing to try was to run CHKDSK with the '/f' (fix any errors) option on the source drive. I did, and there were a few corrections made. However, when I tried to copy that folder again, I got the same error. Uh oh.
After more searching I found a helpful Microsoft article:
It presented several possibilities, but only a few that were relevant to my situation. I was able to rule out 1 possibility - file names that had 'reserved names'. An example would be LPT1 or PRN or COM1. I ruled it out by searching for files/folders with those names in it. Of course nothing came up, but that was one of the simplest checks against the possible relevant issues.
Now things were getting a bit ugly. It was down to 2 possibilities: 1. There are one or more files that contain an 'invalid name', or 2. There are one or more paths to folders/files that exceed the 256 character limit. As far as I know, there are no built-in Windows utilities to check for either of these conditions. After poking around in my problem folder's subfolders, I started noticing that there were a couple of folders that had many, many subfolders. It looked like I found the culprit. This is a good and bad thing. Good because now I know what the problem is, but bad because I'm not sure how to get around it.
A little more searching around on Google for programs that could copy folders without the restriction that Windows itself has found me a possible solution - 7-Zip! It is a program that I already have on this particular server as it is the destination file format (.7z) for the nightly/monthly backups. Upon opening the program and initiating a folder copy of that annoying little (ok, not little - it's about 25 GB) folder, it started copying with no problem! Yaaayyy!
Real quick, I'd like to say that 7-Zip is a great compression/decompression utility with better compression than .zip files, supports a bunch a formats, and is free! Check it out - it's better than the other ZIP/compression utilities that I've tried (including commercial ones).
Well, I hope that this helps someone out there. Feel free to leave any comments/suggestions below.