I did the Camino from StJPdeP to Burgos in Sep 05 and NEVER slept on the floor once. In fact during that season I saw just a few people carrying floor mat; the vast majority did not.
Am starting from Burgos this Sep and DO NOT expect this situation will happen. If it does it will be an experience but refugios, albergues and hostals seem to always have space available at this time of year.
July and August may be different as the number of pilgrims is quite a bit higher than Sep/October.
I just come back from my camino (sjpp-santiago 12/7-4/8 - unbelievable trip), and only one time the hospitalera told me to sleep on the floor (alberque in ferreiros/galicia) because the refugio was overcrowded. in the end I slept outdoor that night.
i think, full alberques were only at the end of the camino, 0 - 100 km before santiago, a problem. but don't be afraid you will allways find a place to spend the night.
Currently in Burgos on day 12 on the Camino having an amazing time. So far there have been no problem with beds, however I noticed that the closer we get to Santiago the more people are joining the Camino and I have also heard that the last 100km is the busiest. I slept outisde one night but only because in Estella there was an excellent fiesta in progress and I didn´t feel like the 10pm curfew. Under the stars in this weather is very refreshing as is the last few days of rain.