To stumble means to nearly fall by tripping or missing a step. When you walk over uneven cobblestones, it’s easy to stumble, so please be careful.
While stumble often means to nearly fall, it can also mean to make a misstep of a different kind. If you stumble over your words, you speak in a stuttering confused manner. Politicians make so many public decisions that they're bound to stumble — make a mistake — occasionally. When you stumble on rare mushrooms, i.e. come across them unexpectedly, be sure have the species verified before you eat them.