Consider the question in class regarding a stone hanging from a thread, with another thread tied and hanging below.
Let downward be positive.

In this analysis, we are asking ourselves what these forces need to be to give the stone a particular aceleration.
-TT + TB + W = ma.

By pulling slowly, we can say that a is approximately zero, so
TT - TB - W = 0.
TT - TB =  W
So TT is always larger than TB and the top thread will break first.

Now, consider what happens when the lower thread is pulled sharply.  We are trying to give the stone a large acceleration downward:
TB - TT=  - W + ma = -mg + ma
If the attempted acceleration is greater than ag, then TB will be greater than TT and the bottom string will break first.