I walked 8 Oct-9November in 2012 and walked a considerable amount in short sleeves. Well, I had the shirt on too, not just the sleeves (hahaha!). Many of the Spanish folks had sweaters on but for me, it was beautiful and balmy. Only once we (my walking buddies, met on the Camino) go to Leon did I find I needed gloves and touque (for all those non-Canadians--woollen hat!), which I purchased for cheap at a China shop. I needed these only for the first hours in the morning, then not really at all after that. We encountered lots of rain, which was warm or not so warm, but never cold like we have in November here in eastern Canada.
One of my Camino friends who walked a few days to a week behind me, after we parted, encountered snow on several occasions.
You never know what you will find, so just be aware that you will be walking in the in between season: just after summer, and just before winter. If you are ready for that and, as C Clearly mentions, enough clothing that you can layer, you'll be fine.
Have fun and Buen Camino!