Why would the arcing of a damaged electrical wire fail to trip a breaker?

Is this a "trip" question?

Don't know. Why don't you ask the electrical forum (https://www.askmehelpdesk.com/electrical-lighting/).

why would the arcing of a damaged electrical wire fail to trip a breaker?
The answer to that question would be yes. The arcing would be caused by a poor connection between two conductors (the wires). The poor connection would be a resistor to the flow of electrical current and would tend limit the flow.

A breaker is tripped when there is an excessive flow of electrical current.