Thanks, your reply led me to look at the layer names - I didn't know layers had names, and the error message made me think the filename was too long.
Looking at layer names, they all look short, but I noticed that layers that were created by the Text Tool do not have a name. I then edited the text within the layer, added some spaces, and it caused the problem during save. If I then removed those spaces, I could save again just fine.
So apparently if you have too much text in a text field, you'll run into this error when you go to save the document. Note that there is no warning about too much text, no mention of text limits, and no mention of text fields in the error message.