The Gita says gahana karmano gatih [4.17] (karma goes very deep). So the concept of Karma is so complicated that hardly anyone can explain all its implications describing what leads to what. But because you already have an idea about the three types of karma, I'll try to explain the question from my understanding.
The thing is, Sanchita Karma is the total stored up karmas whose results have not yet been experienced. So it builds up life after life unless the results of each karma is duely experienced. Whenever there is some karma left out unexperienced in any life, it will get stored up as sanchita karma, later a portion of which is experienced as prarabdha karma. So sanchita karma takes effect or comes into the picture when an individual fails to experience the results of all his actions in his life. Now lets discuss the reasons for it.
The problem is, time of life is limited for everyone. And just like it takes time for a seed to grow up and become a full tree, it takes time for actions to mature and be experienced as results. But it cannot be said for sure whether the person will be alive by the time his actions mature to be experienced. So if he is not alive to experience the results, then it will get added up as Sanchita Karma so that it can be experienced in some later life.
Why doesn't sanchita karma become prarabdha karma?
Well, sanchita karma does become prarabdha karma, but its amount is so big that not all sanchita karmas can be experienced and canceled out in just one life. Hence only a portion of sanchita karma is experienced in one life and that small portion is named as prarabdha karma.
Life is not ordained only by prarabdha karma, the agami karma or kriyamana karma also plays a part. So if for some reason a person commits suicide as a result of kriyamana karma or or dies an untimely death due to some past bad actions as a result of prarabdha karma, then he will certainly have unexperienced karmas of that life which will get stored up as sanchita karma. Hence, even if all sanchita karma becomes prarabdha karma in one life, it cannot be said for sure that all of that karma can be experienced by the person in the same life. So the left out karmas get piled up life after life and become sanchita karma.
Please take a look at the following posts which will also be helpful to know about karma:
Does karma of previous life affect new life?
Is our destiny pre determined? If yes, then why do our actions affect our karma?