Short Story: A secret area full of rare and exclusive Pokémon could be found by accessing the vacant lot behind Bill's Sea Cottage north of Cerulean City
Why We Believed It: A strange gap in the mountains his house rested against could very easily be seen and there was no explanation for why it was there.
Although there were many different stories of made-up Pokémon, very few rumors actually involved accessing new areas in the game. One such rumor though was Bill's Garden, a secret haven for rare Pokémon that, if accessed, would bring the player untold popularity because they were super cool by extension of their rare Pokémon collection.
The rumor started largely because of the gap behind the Sea Cottage Bill resides in. Being at the top of the map with no way of viewing beyond it, players had no idea whether what they saw was as far as it went, or there was a secret land just beyond their sight. Looking at it with a critical eye now, it's easy to see that the sprites required to complete the roof of Bill's house couldn't have been placed atop a mountain texture and so the gap was created, but you miss these things when you're a kid and you're all excited about catching something nobody else has.