Assertions, yes, in so far as they mean "I know this can't happen, but I can't structurally prevent it in the program in an elegant fashion, with the tools I use". However, I can't see how monads are suddenly going to make this problem disappear.

As for throws/try-catch - beyond assertion use, they also report all problems that are outside your control (like IO failing due to HDD damage, or connection dropping). Those kinds of problems are here to stay.