Hi hielienie, whereas the route itself is far less crowded (only 5% of total pilgrims), the albergues can be VERY crowded. This is because there is sometimes only one albergue per stage. Also, some bigger towns that you might expect to find albergues, such as San Sebastian or Ribadesella, don't have them! Hence you might experience the dreaded "bed race" in the summer months. (That said, sometimes the municipality will provide extra mattresses in the albergue, or open the sports hall for pilgrims to sleep in).
I wouldn't worry about it too much, it's difficult to predict, but you might find that you're only affected by the over-crowding a handful of nights. A good trick is to avoid the "official" stages and stop in the small towns or villages in-between. You also need to be a bit flexible and open to staying in non-pilgrim backpacker hostels, pensiones, hostales, or hotels. I have often shared a double/triple room with other pilgrims, and it works out about the same price. There is quite a range of private accommodation options along the Norte, as it's a touristy region, and you're usually able to book something the night or morning before through booking.com.
I walk stretches of the Norte fairly regularly (about twice a year), and I've noticed things getting quiet a lot busier recently, even in the off season. June-August will be busy for obvious reasons, but May and September are also surprisingly busy as it's the most popular walking period for retired folks. The weather is usually still warm in October, so this isn't a bad time to walk. April is also good, although in 2020 it's the Easter period - which might be a bit busier, but you could see some cool Easter parades!