Aside from quantum uncertainty, Yes!.
Uncertainty is not the same as total randomness like people have suggested. Are universe is thankfully governed by laws. We can predict things with a great deal of accuracy. For instance, I can throw a ball and predict with some certainty where it will land, and I know that there will be a full moon on March 2, 2075.
The problem is the more detailed and the further out you want to predict the more you have to know. That would require an infinitely more sophisticated computer. If you really wanted to know everything there is to know, you would need a computer that is at least as large the the entire universe making it impossible.
As far as quantum uncertainty, it does not make as big of a difference as some people would think. The Heisenberg uncertainty principal DOES NOT state that it is impossible to know both the speed and position of a particle. It only says that the more accurately you try to measure one, the less accurate your measure of the other becomes. We can still measure the speed and position of a particle with a great deal of accuracy. Quantum theory relies on probability. We can make positions about particles in atom with incredibly small degrees of uncertainty.
Another way to think about quantum uncertainty is this:
Lets saying you are bouncing a ball off of a wall and catching it. Uncertainty says that there is a small chance that the electrons could be arranged in such a way that the electrons in the ball would not interact with the electrons from the wall, and the ball could pass right though it. However, when does this ever happen? In reality never. Almost everything we experience in our day to day lives is based on Newtonian physics and not on quantum mechanics.
The fact is that advanced computers are able to predict everything from the stock markets to weather using the system you are talking about. The more information the computer has, the more accurate and far reaching predictions about the future are.