How far can computers really predict?

Answer 1

That will vary depending on the real world system being simulated to make the prediction.

Some real world systems are deterministic and simplemaking it easy to make good long term predictions, other systems are perturbed in unpredictable ways making prediction impossible, many real world systems (e.g. the weather) are deterministic but chaotic (i.e. they are very sensitive to undetectably small differences in initial conditions) making them predictable until these tiny differences accumulate to large differences.

For these reasons it is hard to give any simple answer to your question.