In my experience, once a bleeder - always a bleeder. [huh] I'm not saying that all the color will come out of the hat. Just that it's going to take a heck of a lot of wetting to reach the point where it doesn't bleed onto your head or hands anymore. Which is what I'd be trying to avoid to start with.
Yes, I have a brown Dobbs, probably from the 60's. Whenever I get caught in a good rain with it I see brownish drips coming off the brim.
Then the had dries and looks just as good as before.
Never got a stain on my head or hand, but I do see colour in the drips.