The wikipedia says on measurement in quantum mechanics that:
Repeating the same measurement without any evolution of the quantum state will lead to the same result.
On the other hand, doesn't uncertainty (in momentum) entail that I can't expect to measure the same position of a particle twice?
EDIT - I found the following excerpt by Feynman in the second chapter of the first volume of his lectures; it seems related to the question and possibly at odds with some of the answers:
What keeps the electrons from simply falling in? This principle: If they were in the nucleus, we would know their position precisely, and the uncertainty principle would then require that they have a very large (but uncertain) momentum, i.e., a very large kinetic energy. With this energy they would break away from the nucleus. They make a compromise: they leave themselves a little room for this uncertainty and then jiggle with a certain amount of minimum motion in accordance with this rule. (Remember that when a crystal is cooled to absolute zero, we said that the atoms do not stop moving, they still jiggle. Why? If they stopped moving, we would know where they were and that they had zero motion, and that is against the uncertainty principle. We cannot know where they are and how fast they are moving, so they must be continually wiggling in there!)