Well, I put in new needles, checked they were correctly seated, re-threaded, and tested - great. I sewed the hem of my t-shirt. Brilliant! Then I tried to hem the first sleeve, and got right back to the same problems.
One thing I noticed, which seems very odd, is that when it goes wrong it seems that the thread has broken before it gets to the needle, so the needle appears to still be threaded when I remove the fabric, but is not attached to the reel. It isn't shredded, but very neatly broken, and it has happened with both needles, although more often the left hand one.
I have lowered the tension to 3, but if I take it any lower the stitches will be loose (when there are any).
Oh - it occurs to me that I used quilters' tape for the hem, so I'm going to put some on the sleeve hems, just to see if it makes any difference. If it does - goodness knows, because I can't use it on everything I do!!