Well.

Firstly, I find the idea of arbitrary borders as a cause for conflict problematic because all borders are by definition arbitrary. It alone explains nothing. People often cite 'straight line' borders as problematic, yet these exist on the border or the US and Canada, and neither have had many problems in accepting them, as arbitrary as they are. However, one of the fall out issues of Africa's borders is that the central states have until recently been relatively weak (and in many places still are) and as such the borders are arbitrary because of the lack of state-building needed to maintain them as meaningful frontiers, and traditional, more local groupings often intersected by these borders, are a more meaningful centre of power than an often distant and ineffectual state. Essentially then, arbitrary borders aren't the issue, the legitimacy of central government institutions in the eyes of their own people is.

Secondly, Africa has seen innumerable conflicts and movements centred around the break-up of states for a variety of reasons. The most recent example is South Sudan splitting from Sudan in 2011, however the current situation in DRC is essentially a mosaic of different local movements that totally control small areas and fight each other for control of more. On paper it looks like one giant country, but in reality the governments areas of influence and quite limited. So as a direct answer to your headline question - there are plenty of examples of just this happening.

It's worth noting that no central government will allow part of their dominion to break away, because it means less income, less taxes, less people and less power, both domestically and internationally.

Tied to this, many civil wars from regionalist or ethno-centrist groups want to take control of the state, not break away from it. Taking control of a state, larger than their 'home land' gives them access to the income and people not just of the region they're loyal too, but also the income and people of their rivals regions as well, which means they can better protect their home region and privileged position at the top of power.

Imagine it in a familiar, hypothetical concept. You are a Scottish separatist, you feel that Scotland is being undermined or directly mistreated by the central British government. Times are tough and things are desperate and you raise an army. Because the British government has been bad at building a centralised state, it can only muster an army half the size of yours. You defeat this army and it scatters, do you turn around secede and head a new government that rules around 3 million people in your homeland, or do you march on London unopposed and take control of the government, ruling 65 million people, including your homeland?