Shifting gears in a bus is not permitted when:

Shifting gears in a bus is not permitted when crossing railroad tracks due to safety considerations. When a bus approaches railroad tracks, it's crucial for the driver to maintain control and stability. Shifting gears at that moment could lead to unintended speed changes or loss of power, which may compromise the driver's ability to safely navigate across the tracks. Ensuring that the vehicle is in the appropriate gear beforehand allows the driver to focus on the crossing without the distractions or risks associated with changing gears. This practice helps maintain awareness of the surroundings and prepares the bus to react swiftly if a train is approaching.

In terms of the other scenarios, while driving down a hill or crossing a bridge may require careful attention to speed and handling, the specific danger posed by shifting gears at railroad crossings makes this situation particularly critical from a safety perspective. This is why the correct answer emphasizes the importance of remaining in a stable gear while negotiating railroad tracks.