To assume each branch of government will "go to war" is simply a misunderstanding of parliamentarism ("Parliament is supreme").
If OP did not mean, literally, the branches fight each other physically, but a more modest "in conflict" -- whereby the legislature's intention was not interpreted by the judiciary appropriately, and in similiar vein, the government did not enforce laws consistent with intent of legislature -- the answer would of course have to be yes.
Conflicts Between Each Branch of Government
There have been conflicts between each branch of government. How often are they in conflict? Almost every single (working) day -- just watch the evening news of most liberal democracies ("democracy", of course, is the other key pillar without which seperation of power is meaningless).
Do branches of government actually function as per the trias politica philosophy whereby there has to be a strict separation between three independent powers in every nation? Of course not.
Montesquieu's The Spirit of the Laws, published mid-18th century, which inspired both the French and the American intellectuals during their revolution, is still merely theory. Very good political theory albeit. To my knowledge, there is no single accepted indicator of a nation's perfect implementation of this model.
So, conflicts between branches happen often. If it deteriorates further it would be called "civil war" and the end of democratic institutions because martial law would be imposed to restore order, failing which a new nation will be formed. A successful imposition of martial law is known by another name, "coup d'état". That ends a democratically elected (legally derived powers of) government. How many martial law regimes have there been? Well, we can count that (I haven't).
Montesquieu's The Spirit of the Laws: In Practice
How do branches actually work in daily life, if it approximates, but not achieve a clear separation of powers? Too much for me to explain all the details, and I'm not sure I'm capable of it. In any case, this was answered by Bagehot in his book, "The English Constitution" (1865). This book was for the British system of government (parliamentary democracy, whereby the head of state is a distinct person from the head of government).
For America, it would be presidential system, whereby the head of state is the head of government. And Bagehot's contemporary, Woodrow Wilson, the 28th president of the United States, wrote "Congressional Government: A Study in American Politics" (1885).
In sum, how do democratically elected branches of governemnt function imperfectly (according to the Trias Politica model)? Both Bagehot and Wilson answered this in their respective 19th-century books.
(I suppose this is a historical answer).