Yes, it's quite common as the United States is what's called a "Federal" nation.
A Federation is an organization of like minded soveriegn regions (in the case of the U.S., the states and incorperated territories) band together and under an agreement (U.S. Constitution) decide to cede some but not all rights of soveriegnty to a created central government. Which rights are ceded are based on each Federation's agreements. For example, in the United States, immigration is a right ceded to the federal government and all states and territories lose their right to determine who can stay and who can go once the Feds say they can enter. The Swiss Confederacy* reserves this right for the individual Cantons (State Equivelent) to decide and once one Canton declares a person a citizen, they are citizens of Switzerland and thus can move to other Cantons.
This organization of regional government means that the member states of Federal Nations do retain some rights traditionally held by the governments of Unitary States. Where legal authority is not delegated to the federal government to make the laws, it falls to the member states to make and enforce those laws (The 10th Amendment of the U.S. Constitution makes this explicitly clear: The Federal Government can only make and enforce laws that the Constitution explicitly says they can and where the Constitution does not grant the right to the Feds, that right defaults to the state. The U.S. Constitution has Thrice been amended to change what the Feds could make legislation about: The 13th Amendment moved all laws with respect to non-punative slavery (i.e. forced labor as part of criminal's punishment) to the Federal Government (and banned it just to be clear on the matter), The 18th Amedment famoulsy moved alcohol regulation to perview of Congress even if the manufacture was entirely intrastate (and banned it, just to be clear on the matter) but the Volstead Act had to be passed to actually enforce the ban. Finally, the 21st Amendment gave the right to regulate alchohol back to the states, which meant the Volstead Act was unconstitutional from that point on (and we've been drinking ever since.).
Similar countries to the United States that are both Federal Nations and Common Law Nations include India, Canada, and Australia, Malaysia (notably the only Federal Monarchy in the world! Canada and Austrilai don't count, as while they are both Constitutional Monarchies (they share a monarch with the UK) their individual Member States are not monarchies. Malaysia elects it's king to a five year term from a pool of 9 heredity rulers of the member states), Micronesia, Nigeria, Saint Kitt's and Nevis, Pakistan.
The United Kingdom is a Devolution (oposite of a Federal nation... where the central government chooses to allow regions to be semi-soveriegn but can reabsorb them if they so choose), with Scotland and Northern Ireland having their own Judiciaries (England and Wales share the same Judiciary).
Nations that are Federations but not Common Law Nations include Argentina, Brazil, Mexico (all three structured very similarly to the United States), Venezula (and formerly other members of the Gran Colombia period of the region's history, though Gran Colombia broke up and the Federation of Colombia became a unitary state), Germany, Austria, and Switzerland, Russia, Belgium, Ethiopia, Nepal (the newest Federation, est 2015), The United Arab Emerates, Iraq (following the Iraq war).
As the United States is the oldest extant Federation in the World (though the first "federal" nation of known to the world and the longest standing federation from formation to collapse goes to the Inca Empire, which existed as a federal government from 1197-1572. Though it was far more likely that the Irish Confederation and Federated Dutch Republic were more significant influences on the U.S. federal system) the United States serves as a good template for most Federal Nations organized today as it was super easy to copy the U.S. constitution and adapt it to the newer nation's unique needs.
*Confederacy here uses the old meaning of the word, which is what we today call a Federation. Confederacies today refer to governments where the member state is equally soveriegn to the created central government... the general litmus test is if there is codified laws that say that once a member state joins, they can choose to leave. The closest government to a real Confederacy is the European Union, which as we are now aware, is famoulsy exitable. The U.S.'s bloodiest war in it's entire history resolved that this is not a right of the states.