The filter I use can. What you would be filtering out is radioactive DUST.
As the man said, quality of rainwater depends on what is upwind from you, but is MOST cases, it's fine (naturally distilled, y'know) If you are just downwind from the company that makes isobutyl bad stuff, or methyl ethyl ugly, maybe not.
But look at it this way- after the Apocalypse, industrial pollution will be gone. And coal and charcoal are two very different things.
What we have here is... failure- to communicate.