I can't imagine how anything but a dual courts system would work in the United States. There are state laws and there are federal laws, and the appropriate court systems are necessary to see those are enforced.
The system works very well, but there have been some conflicts from time to time over which law -- state or federal -- applies to a case. There is a whole body of case law covering such conflicts of law instances and it is an ever evolving field. Whether federal or state laws control in a given case is usually a cut and dried matter, but not always and it is in those rare instances when it must be determined which laws reign supreme.