At the beginning of the game, you see Crono
woken up by his mom in their house, before he goes to the Millenial Fair. In the Japanese version of the game, this chapter is called "The Journey Begins!
Dream of the Millenium Festival" (instead of just "The Millenial Fair").
After defeating Magus, you see Crono
woken up by Marle in their house... before he
wakes up "for real" in Ayla's Hut.
Then you learn much later that the whole adventure is like, the
dream of the planet...
I believe the game (the whole series actually) just wants to mess up dreams and reality, to show that dreams can come true and reality can be like a dream yada yada...
Just my personal interpretation