on the brink of / on the verge of / on the cusp
on the brink of / on the verge of / on the cusp

Is there any difference between these words?

on the brink of
1) There is no evidence that the Iranian regime was on the brink of collapse or that they were forcing the termination of Iran’s civilian nuclear program.
on the verge of
2) The agreement was so poorly negotiated that even if Iran fully complies, the regime can still be on the verge of a nuclear breakout in just a short period of time.
on the cusp
3) The agreement was so poorly negotiated that even if Iran fully complies, the regime can still be on the verge of a nuclear breakout in just a short period of time.

Question

on the brink of / on the verge of / on the cusp
on the brink of / on the verge of / on the cusp

Is there any difference between these words?

on the brink of
1) There is no evidence that the Iranian regime was on the brink of collapse or that they were forcing the termination of Iran’s civilian nuclear program.
on the verge of
2) The agreement was so poorly negotiated that even if Iran fully complies, the regime can still be on the verge of a nuclear breakout in just a short period of time.
on the cusp
3) The agreement was so poorly negotiated that even if Iran fully complies, the regime can still be on the verge of a nuclear breakout in just a short period of time.

Answer

Just a few random thoughts:

"on the brink" is usually used for something negative and fairly important: on the brink of collapse/disaster/catastrophe. The image is one of standing on the edge an abyss.

"on the verge" is often used when you almost do something e.g. "I was on the verge of cancelling everything, but changed my mind."

"on the cusp" is used for boundaries between two states of being. A teenager is on the cusp of adulthood, for example.

I wouldn't say that they are interchangeable. We use them in different contexts and collocations.

Answer

They all mean the same thing, and can be used quite interchangeably - both for negative and positive events.  On the brink, verge, cusp of success or failure.  In terms of how you actually see these terms used, 'on the brink' is very often seen in the context of war for example - on the brink of war, etc.  But again, you can substitute - verge of war, cusp of war too.

Answer

refer to SU.ki's answer because they are used in different contexts exactly as su.ki has explained