If you say that someone is falling apart, you mean that they are becoming emotionally disturbed and are unable to think calmly or to deal with the difficult or unpleasant situation that they are in. I was falling apart. I wasn't getting any sleep. = crack up
If an organization or system falls apart, it becomes disorganized or unable to work effectively, or breaks up into its different parts. Europe's monetary system is falling apart I've tried everything to stop our marriage falling apart. = break down
[ 'fol ] (verb.) before 12th century. Middle English, from Old English feallan; akin to Old High German fallan to fall and perhaps to Lithuanian pulti.