I walked the Camino Frances through June and the beginning of July this year. We had no trouble finding beds in Spain. Only on a few occasions did I see people miss out on beds (O'Cebreiro was one such place) but these people usually arrived very late afternoon or evening.
The first day out of Sarria came as a bit of a shock to me, as there seemed to be 'hundreds' on the trail. The trail is certainly busier in the last 100km, but I think the fact it was a Sunday may have had some bearing.
I thought before I arrived in Spain that I would find it hard to stay in big dormitories in crowded refugios. And some refugios certainly have the beds crammed together. However, I found I was actually quite happy staying in these places, as it meant I was able to have more contact with people I had met along the way. I just bought earplugs in the pharmacy in Viana, which had a great selection... and that kept me sane at night ;-)
Margaret