You are required to register your child for participation on the first day. That involves filling out a medical form (allergies, etc.) and a basic permission slip. My understanding agrees with Kathy--no child who is signed up on the first day is ever turned away. On cruises that had relatively few children, I've seen them allow registration thru the whole cruise. On other cruises, I've seen them close the registration after the first day. I don't know if they would allow a late registration if you asked.
However, they plan on the ages of the kids when they take the bookings. They can shift counsellors among the various age groups to provide the needed staff in each group. For instance, we frequently sail during the school year. At these times, there seem to be more counsellors in the preschool areas and fewer with the older kids. There is some fine print about numbers and safety, but I've never been aware of a child being turned away for that reason.
There is also fine print about behavior, etc. My understanding is that this is rarely needed, but can be enforced if a child is totally acting out.
And, yes, the number of kids on board is always a surprise! It never feels anywhere near the number that they always seem to have.
Kathryn
