When and why do individuals and groups stop believing in a political process and turn to terror and violence to achieve what they see as justice?

Individuals and groups may resort to violence when they realize that the government or the administration will not make any concessions towards their demands. Noticeable breakdowns in the political process lead to loss of confidence among the people. For instance, political processes are seen to be breaking down when the governing institution imposes itself on the governed through force, coercion, and intimidation. In such governments, the leadership would want to increase administrative powers or extend leadership terms against the will of the people. In such a situation, public protests would ensue to fight against the undemocratic practices of the leadership. The protests might start off peacefully but turn violent, forcing the government to respond through police action. A meeting of the protesters and the police on the streets can increase the chance of violence between the two groups.
A real-life example of such a situation occurred in Ukraine in 2013. President Viktor Yanukovich failed to pay attention to the demands of the people through the opposition. The opposition proposed a reduction of presidential powers, constitutional reforms, and changes in the governing system.  Yanukovich rejected the proposal and held on to the excessive presidential powers. He stifled parliamentary processes to pass amendments by coercing and intimidating the members of parliament. The stalemate led to the spread of violence from Kiev to the rest of Ukraine. The situation led to his self-imposed exile to Russia and ouster as president of Ukraine.
http://www.lse.ac.uk/international-development/conflict-and-civil-society/past-programmes/csrc-publications-resources

https://www.aljazeera.com/indepth/opinion/2014/02/violence-death-failed-politica-2014220153327383148.html

https://www.rt.com/news/ukraine-kiev-death-toll-955/