Error 1322.A portion of the path exceeds the length allowed by the system.After trying the installer from the root of the C: drive didn't solve the problem, I installed the Microsoft Fix-It, which is an update to the venerable Windows Installer Cleanup Utility. That didn't solve the problem either. I thought it was an issue with that particular machine, but after trying it on another machine (and another version of Windows) I concluded that something must be wrong with the installer.
I searched Google and LibreOffice's bug tracker. Amazingly, I wasn't finding anyone else who had this problem. I even computed the SHA1 hash on the installer on my flash drive and compared it to the SHA1 hash of the installer as downloaded on my hard drive, to make sure the flash drive wasn't corrupting my files.
Finally, I went into μTorrent on my main PC to force it to re-verify the original download. For some reason the program wasn't running. When I fired it up, μTorrent resumed downloading the installer at 81%. Turns out, Sue closed μTorrent because it was interfering with her Hulu streaming.