Whether or not you carry a sleeping bag depends on the time of year, I think.
We traveled beginning in SJPP around the first of September and Joe carried a fleece liner. Very lightweight and proved to be perfect. I carried a Marmot Pounder Plus Sleeping Bag (1.5 pounds) and it was also perfect for me, but I tend to get cold easier than he.
If you travel in the summer months, some people simply take a silk liner.
If you travel in winter, you will definitely need a sleeping bag.
As September goes on, it gets colder, especially at night & in the mornings. Most albergues don't have blankets, so you'll need at least a light-weight sleeping bag. If it's warm enough, you can sleep on top of it.
Peaceable Projects Inc. is a U.S.-based non-profit group that brings the vast resources of the wide world together with the ongoing needs of the people who live, work, and travel on the Camino de Santiago pilgrim trail network in Spain.
My wife and I did the Camino is September, and we did not carry a sleeping bag. A fleece bag liner and a silk sleep sack was it. Remember, you will be staying indoors, with a number of close friends whose 98 degree F bodies will bring warmth to the room, even if the room is not heated.
The fleece liner was rated to 55 F and the sleep sack was good for another 10-15 degrees F lower.
Also all the albuges we stayed in did have blankets but the number may be limited, most bunks had blankets on them however.
I am finding this question a very tricky one to answer ... I went March-April, taking a verr light bag but I was only inside it maybe three or four times .. the rest of the time I slept on top of it .. I am naturally warm .. I sort of generate heat (I went over the Napoleon pass in snow in shorts!) ... I do like to have my feet out and this bag didn't have a two-way zip so it was always too hot for me. In France there were always blankets and in most of the refuges in Spain I found the same (I didn't use them, just noticed).
Thing is - there is this fairly unpredictable recent changeable climate ... then low vallies and high mountains .. then arid zones, and wet zones .. how on earth does one predict what to take? and I want to do some wild camping this year too!
.. so, after listening to others I am taking a 700gms comfort 7c low 2c, bag plus a set of long-sleeved thermal underwear and a woolly hat for if I need to have another layer (thermals tip via Sil) ... and can wear them too whilst walking if it all goes pear.
The weather in April and May can be tricky - as it can September and October I imagine. I have started April 8th, 13th and 20th from SJPP. The year I started on the 20th was the coldest (2007). It snowed lightly walking out of Leon in May. Two or three days later the sun was scorching hot, and many unprepared pilgrims ended up with really bad sunburns. It's spring and not too predictable.
I carry a sleeping bag that weighs less than a kilo with a silk liner so I can wash it. Sometimes I was too cold and needed an extra blanket, sometimes I was too hot and just used the liner.
The only thing about spring weather is that you have to be prepared for both heat and cold - but the spring flowers make up for any inconvenience! Buen camino