I've been a fan since 1960, having grown up in North Tonawanda. So the four super bowls were not a factor in my fandom, except for the disappointment.
Just to set the record straight, the Brown won four AAFL championships and, since joining the NFL, they have won four NFL championships. The last NFL championship was in 1964, the same year the Bills won their first AFL championship. (That would have been a great superbowl.)