For some reason, all of the newer torrents I had added would start, and then after a few seconds they would automatically pause. Each one said Error: Permission Denied. Every time I restarted them, they would pause again.
My first thought was this was a tracker issue, and I needed to be registered with them. However, this was happening with a bunch of torrents, and they were downloaded from sites that I've never had a problem with before. I had 2 torrents that were still functioning just fine. I searched the Torrent site's forums, and then searched the Transmission forums, but there was no answer.
I didn't realize at that time that this was only occurring with the newer torrents.
After a few days, for unrelated reasons, I was messing with the permissions for my external drives. I set them all to ignore ownership on this volume. When I opened Transmission and hit Resume, all the torrents began downloading normally.
After thinking about it, I remembered I had changed the permissions a few days before, as I was diagnosing a system issue.So the entire problem was a simple permissions issue.
Transmission was being denied access to to its Download folder on one of my external drives. Only the torrents that had been in progress before I had changed the permissions the first time were working properly.