I tried something new and it seemed to work for me this time. So now, I think I have some more information that gets me closer to understanding the issue we are seeing.
Our packaging team has recently taken over the packaging of Adobe Reader and I noticed that they have a different AIPs for each version 11.0.x. When I was testing, I previously uninstalled Reader, and reinstalled it from the 11.0.01 slip-streamed AIP that they created (not thinking that it would make a difference) before trying to install 11.0.02 and 11.0.03.
This time, I completely uninstalled Adobe Reader and reinstalled the base install (11.0.0) and then installed all the updates in order, and this time it worked as it should.
So... I believe the reason it failed was that I was trying to apply the update to a client that had a "slip-streamed" installation. If the client started out with the base 11.0.0 installation to begin with, then I'm thinking we shouldn't encounter the issue.
I'm still a little currious as to why the updates won't actually update a computer with a slipstreamed Reader installation (assuming that is the issue).