It's not a firewall mishap, there have been no changes to the system, and I can verify the connection is going out like it should. It seems like it's either a server side thing, or a temporary network connection problem.
Note that the error dialog is different if Live can't connect to the server at all. Here's the error if I deliberately block access: "Live could not connect to ableton.com"

But this is the error I'm currently having a problem with: "Secure connection to server failed"

This happens even if I completely disable the firewall and make sure absolutely all network traffic goes out, from anything. Also, I repeat, it worked for days completely without a hitch.
The way things are going, though, I'm quite sure I'm going to be updating to a full license in the next couple of days