Well wouldn't you know it, not 2 minutes after I started down that path I ran out of bobbin thread (no this isn't the issue). I re-threaded, popped the bobbin back in, and now it won't catch the bobbin thread. No, I didn't have the bobbin in backwards, but just in case I did I put it in the other way, and to make doubly sure it wasn't an issue with the thread I was using I tried another bobbin with no luck. I even went so far as to clean the machine and still no joy. The needle is in correctly, and the tension is good (it was working 10 seconds before this). Of course I don't have a manual for the machine and it's not like I really paid attention to what was going on down there when it was working!
Arg! There go my best laid plans.
ETA: Apparently, unlike my computers, my sewing machine responds to shaken fists and muttered curses! I have now managed to complete the projects I was working on, stitches turning out beautifully.