Is it possible for a country to have a mixture of more than just one type of government?

A mixed government is indeed possible. It also may be called a mixed constitution or a balanced government. Nevertheless, by definition, mixed government combines elements of democracy, aristocracy, and monarchy. Thus, it embraces the views, desires, and impact of the many (democracy) with those of the few (aristocracy) and a singular person's (monarchy). Because of the structure of a mixed government, issues are decided in a variety of ways as well. For instance, democracy requires participation of the people (demos), so some decisions are made by popular vote with majority rule. Aristocracy implies that some decisions are made by the few (aristocracy). Then lastly, the one (monarch) makes some decisions. In the English form of government, under which the British North American Colonies resided prior to the War of Independence, the House of Commons, the House of Lords, and the Monarchy (a mixed/balanced government) addressed the “the differing ‘Ranks and Interests’ of the society at large” (The Founders Constitution).

When studying the Enlightenment, Montesquieu’s ideas about balanced government may be the first political theory presented in this regard. Classical writers like Plato, Aristotle, Polybius, and Cicero also believed a balanced/ mixed government was better than dictatorship or absolute monarchy. Also, James Harrington, who wrote during Oliver Cromwell’s Commonwealth period in England, championed the idea of a mixed government with popular sovereignty. During the Renaissance, Niccolo Machiavelli reiterated the classical basis of mixed government in his works. 

Montesquieu used the British government as an example of balanced government, but made no distinction between mixed/ balanced government and the separation of powers. James Harrington’s idea of “a system whereby one part debates and divides, yet another part resolves or chooses, and still another executes” was more influential in American government than Montesquieu's writings (The Founders Constitution). The United States Constitution's institutional checks and balances were the result of creatively tweaking the classical and enlightenment ideas of balanced/ mixed government. The outcome: three separate but equal branches (legislature, judiciary, and executive) whose powers checked and balanced one another became another iteration of mixed government.