The common denominator in both instances is Firs, the old and loyal former servant to the family.  He says in Act II that the last time he heard this sound was just before the servants were freed -- a huge transition in the Russian social order.  This indicates that another huge transition is on the horizon for the Russian people -- and so it is.  The Revolution will change everything about the country.  At the end of the play, Firs says:

My life's gone just as if I'd never lived....I'll lie down a bit.  No strength left.  Nothing's left.  Nothing.

And with these final words, "the mysterious sound of a breaking string" is heard again, followed by the sound of the axe falling on the cherry orchard.  Both are symbols of a society whose way of life is on the brink of a complete transformation.  Is the transformation destructive or productive?  Well, to judge by the characters in this play, it seems to be destroying everything that most all of the characters have ever counted on.

The fact that Chekhov identifies this transition with "the sound of a breaking string" calls to mind the metaphor of "hanging by a thread."  This metaphor suggests someone (or a group of someones) who is at his/her final grasp and maintaining a very precarious hold on something that he/she cannot seem to let go of, no matter  with how much stubbornness he/she ignores the fragile, breakable nature of the thread.  This, again, typifies many of the characters in The Cherry Orchard, characters who insist upon living in the "good old days," rather than face the harsh realities of the present.

For more on the "breaking string" and the historical context of the play, please follow the links below.