Unlike dehydration, oxidation only affects one carbon atom.
In oxidation, the atoms removed from an alcohol molecule are the hydrogen of the OH group and a hydrogen atom from the carbon atom joined to the OH group.
Only primary and secondary alcohols can be oxidised in this way, because they have a hydrogen atom attached to the carbon atom which is bonded to the alcohol (OH) functional group. Tertiary alcohols, however, do not have this hydrogen atom.