I finally got a model that would converge in RSiena. My plan was to start with a simple model, and then to add more of my hypotheses once I got the simple model working. It turned out to be tough enough to get the simple model to converge that I just went with that once I had it working.
At Sunbelt, I went to a great workshop by Tom Snijders, which was very helpful. He also came to the poster session, and gave me some great tips about how to work with my model.