are there levels of nut allergies?

My daughter is in 3 year old kindy and the schools policies is not nuts. When I asked the teacher why she said that there are 3 kids in her class alone (20 kids in all) that have a nut allergy. Please forgive my ignorace on this matter but how many kids do actually have nut allergies and how many parents just basically overreact. I understand the severity of nut allergies, but when I asked my daughters teacher what products are we not to bring to school she just said Oh only a ban on nuts, food products with traces of nuts are fine just the kids in your daughters class are only allergic to whole nuts..the question im asking is, I thought if you had a nut allergy then how can you be allergic to nuts but not products with traces of nut, and if these kids are apparently "anaphylaxis" then how can they be fine around products that have traces of nuts, it makes no sense and a little bit annoying that my daughter cant have her favourite nut bars to eat at school becuase these kids are apparenlty Highly allergic.