Yes. Think Apple. They weren't the first Apple, and they can't touch other Apple entities that have been around before the existence of Apple (ie. Apple Bank in NY). However, anyone that uses the name Apple, or a logo similar to that of Apple's, after Apple's original use, they can go after.
Side Note: trademark protection is more focused on protecting consumers than companies - unless the company has spent an excessive amount promoting and selling their products, at which point, they are given special protection/rights.
Side Note: trademark protection is more focused on protecting consumers than companies - unless the company has spent an excessive amount promoting and selling their products, at which point, they are given special protection/rights.