Do cats have colds

Healthy cats have a warm, dry nose.

Cats' noses are generally supposed to be cold and wet. If their noses are warm and dry, then there might be a problem. not the Word "might". It is not necessary for them to be sick if their noses are not wet and cold. Only if they show other symptoms of sickness you should be worried.

But normally, because cats lick their noses, when healthy, their noses are wet and cold.