When you walk 780km across a country, through 5 different regions, 350 villages and towns, you are going to hit the occasional 'busy' time somewhere along the way!
If it isn't starting off with a big group (some days in May over 200 pilgrims a day leave from St Jean Pied de Port) it could be finding accommodation a problem because it is harvest week in Navarra, or a big fiesta somewhere else.
Last year I added a post to my blog on Spanish Fiestas.
http://amawalker.blogspot.com/2012/08/s ... amino.html
"Over 15 000 fiestas and festivals are held in Spanish villages and towns throughout the year. Many are religious, celebrating local saints, some are Regional and others are National events such as Semana Santa (Easter), Corpus Christi and Christmas. Prohibited during the forty years of the Franco era Carnaval is celebrated in many Spanish towns a week before Ash Wednesday."
The Camino is dynamic - there isn't one 'busy' time that can be avoided along its entire route and then the rest of the walk is solitary. (Perhaps only the month of August could be avoided). Wherever people start walking from - Pamplona, Burgos, Leon, Astorga, O Cebreiro etc, you'll become a part of the ebb and flow.
Seek out albergues in the smaller villages, in-between the stages mentioned in most guide books, and you should find fewer peregrinos queuing up for beds.
Just go with the flow! Enjoy the busy times, they were there in the Middle Ages and we haven't nearly caught up to the multitudes that walked then, and savour the moments when you have time for contemplation and reflection.