How Trustworthy is Science?

Any student of science knows that the first lesson is in scientific method. That this enables us to put our trust in science. Scientific method leads on the idea  of reproducibility. This means that an idea or theory can only be accepted as true if it is proven through experimental means, and those results have to be reproducible, i.e. someone repeating your experiment will get the same result. However a study found that only 25% of papers were reproducible. Why is this the case? And what does it mean for science?