I will pick 2 examples however I don't have any makoros at the moment (so please help to provide or correct me/help edit.)
If a katan (child) is not neman (believed) in regards to tovelling celim for someone. Would he be believed in regards to using celim (vessels) that he tovelled. That is if his parents came along and saw him eating from a certain cli and the parent asked "was that tovelled" and they responded "yes" do we believe the katan? Or is it better to give him a cli that we for sure know was tovelled?
What about if the katan was eating food that need to be checked for bugs and we asked him "did you check that food" and he respods "yes". Can we believe him or better to give him food that we know for sure was checked.