So I tried again this evening, to see if I could sew across the small test quilt without breaking the thread. The first thing I managed to do was jam the needle. After getting it unjammed, in trying to get the thread out of the bobbin area, I managed to get it ALL apart. Which meant I had to get it back together again. It seemed easy, but I just couldn't do it. So DH had a look at it - the only part I had missed was getting two small levers snapped into position. Everything else was okay. So, I tried again - and this time - success! I managed a little bit of stippling all the way across, and the thread didn't break!
Now on to advancing the quilt a bit and practising some more. It's a baby step, but it's a step in the right direction!