Can a baby be allergic to some brand of diapers?

Yes! Some diapers get that white color because they are treated with chemicals that may leave dioxins on the diaper and these may cause rashes. The other culprits are baby wipes also due to chemicals and diapers left on to long causing dermatitis on a baby´s skin. Consider trying cloth diapers or switching to a "greener" brand of disposables.
Contact allergy. Yes they can. Baby's skin is very thin and sensitive. Sometimes the scent or certain chemicals in a particular brand of diaper could be causing the reaction. U want to consult ur doctor or an allergist. Of course it can always be a diaper rash.