When the party reaches the Lonely Mountain at sunset on Durin's Day (the last full moon of autumn), they search for the hidden keyhole by the light of the setting sun. When the sun goes down and the light is lost, they give up hope and start back down; Bilbo remains behind, and sees the keyhole by the light of the moon.

However, a full moon at sunset can only be a rising moon, hence the moon would be on the other (east) side of the mountain and unable to cast its light upon the keyhole.