Once a user closes out of the Emergency Bulletin for the Lightbox/Banner they will not be prompted with the Emergency Bulletin again until they clear their cache/cookies or open up an incognito/private browsing window.
When you clear the light box or the banner emergency bulletin a cookie is written to the browser cache to notify the site that the emergency bulletin has already been cleared.
This will prevent the emergency bulletin from coming back up as long as that cookie still exists. That means it will not come back in that particular browser until the cookie/cache is cleared.
The settings for this completely depend on the browser and the user. Some users have their browser set up to clear cookies every time they close it, whereas some people don't have it clear except through a manual process.