These are metaphors for common events.

You're working for a client who wants some changes, and more changes, and more changes, and is difficult to work with. At some point you may have to fire the client. As a simile, this is like a rubber band being stretched to the breaking point.

Or, the new boss is a very bad manager. Some people quit to move to another job. Each one is a rope snapping. If enough ropes snap - if enough people quit and are not replaced fast enough - then that department can no longer function.

To clarify further, "rubber band to break" is because a rubber band can be stretched and stretched, but not without limit. Once it has reached its limit, it breaks with a snap, and cannot easily be fixed.

For the metaphor "one of the ropes to snap", think of a cargo net carrying a heavy load. Each rope has a different tensile strength, and the load is not perfectly balanced. It's possible for a heavy load to break one rope, but the net still hold because the load shifts onto the other ropes, which have reserve strength. However, this should be a warning, because as more ropes break the less reserve there is, until the net is no longer able to hold the cargo.

I don't know what you mean by "eventually will go wrong" in this context where several senators say that things have already gone wrong.