Technically, the actual whole point of wrapping is to avoid leaking implementation details. If you let "FooLibraryException" bubble up, and then you stop using Foo Library, then all of the users of your code are going to end up broken waiting for "FooLibraryException" when now you throw "BarLibraryException". This diminishes any value exception handlers theoretically could provide since you end up having to wrap everything at each step anyway.

Checked exceptions were introduced to try to help with that problem, giving you at least a compiler error if an implementation changed from underneath you. But that comes with its own set of problems and at this point most consider it to be a bad idea.

Of course, many just throw caution to the wind and don't consider the future, believing they'll have moved on by then and it will be the next programmer's problem. Given the context of discussion, we have assumed that is the case.