You need to find out 1) whether you and the other company are in the same line of business, and 2) who used that mark first. Regardless how you acquired the name, if you are infringing, you would probably expect a cease and desist letter from them.
Laches is an equitable defense to trademark infringement, which would make the whole case tricky. But laches from the other company does not make your infringement go away unpunished. In California, for example, an aggrieved party has 4 years to enforce its trademark and that is the statute of limitation in law. Laches can happen earlier but it must be determined case by case by an attorney.
