I walked the CdN in 2016, starting from Irun on the 5th of May. No problems finding a place to sleep, except for 1 night: the first stage out of Bilbao.
Apparantly there were the pilgrims who started in Irun, accompanied by pilgrims who flew in on Bilbao airport, starting their Camino from there. I got into the albergue in Portugalete, bus shortly after that it was completely full. Pilgrims had to rely on staying in -more expensive- private lodgings. And the hospitalero of the albergue granted pilgrims to sleep on the church porch (in the open air, but at least with a roof over their heads), providing them with extra blankets.
Other than this: no problems whatsoever. Always stayed in municipal albergues (or ones run by foundations) when there was one, or private albergues. Never had to stay in a pension, casa rural or hotel.
Luckily I was able to walk this Camino the way I like it: starting to walk in the morning without having decided where I to stop that afternoon.