The studentsʼ picnic plans __________ when it started raining heavily in the morning.

The sentence describes how the students' picnic plans were ruined or cancelled because of heavy rain in the morning. It implies that the rain disrupted their arrangements, making the picnic impossible. The blank needs a phrasal verb that means that something "failed to happen" or " was unexpectedly cancelled". 

"Fell through" (option D) is a literal fit here, as it literally means "collapsed", or "failed to materialise". 

"Dropped out" (option A) means to quit, "dropped off" (option B) means to deliver/decline based on the context, and " fell behind"( option C) means to lag; none of which are analogous to what the sentence demands.

The correct choice is therefore option D.