L.M.
The short answer is yes, it can come on little ones very quickly. My example is my daughter who has allergies (hasn't been tested yet). Fine on Sunday, started sounding congested on Monday (clear discharge only and no fever). Tuesday more of the same but coughing when we picked up from preschool. Wednesday morning to the doctor we went, they are thinking bronchitis or pnemonia (yes that quick). Xray says not pnemonia but doctor feels it is so it is called "clinical pnemonia" and still no fever. Thursday morning, low grade temp that went away. Still bad on Friday so breathing treatments and a second antibotic. That was 10 days ago and she is still coughing some and has some congestion but it is clear and no fever.
So she missed preschool Wednesday through Friday, home through the weekend, and off on Monday for MLK. It was a big inconvenience and my hubby and I had to tag team the shifts but she was well when she went back, taken care of while sick, and not exposing others.
If your child has yellow or green snot coming from her nose and has a fever she is contagious and you have now exposed the kids from her daycare as well as her teachers. That is how the whole class gets sick and some get reinfected after getting better. She should be fever free without Motrin or Tylenol for at least 24 hours before she should be allowed in a daycare setting.
Also, the less she is exposed to cold air the better she will be.