The scientific method allows us to make a model of reality which appears to work to a great degree of accuracy. There are, however, two major limitations: we can only model what occurs when there are many observable occurrences; and we can only precisely model what will occur when there are finite possibilities. This means we cannot answer questions such as: "What am I?"; "Was there existence before time?"; "Where, exactly, is a given particle in phase-space?"; and "Is this painting good?".
The first limitation seems straight-forward and generally the methods humanity follow here are alternative belief systems ...from the religious belief, "God was before time."; through the subjective belief "I think this painting is one of the best I've ever seen."; to the logical belief "I am a complex system of self references that happens to produce an entity capable of pondering such things.".
The second limitation seems at first to pose little threat, after all when are there infinite possibilities? However it seems that quantum physics gets limitations here (exposed as the uncertainty principle). A mathematical interpretation of the likelihood of unobservable states is that of a negative probability; this is akin to defining a region of phase-space in which the particle in question could be anywhere-when.
All our experimental methodology is based on the principle of the statistical stabilization (the law of large numbers). All experiments are prepared in such circumstances that relative frequencies must stabilize. This is the result of our cognitive evolution. In the process of evolution the brain extracted from the chaotic and (lawless) reality phenomena which satisfy the principle of the statistical stabilization (repeatability in the average). These and only these phenomena are considered by the brain as real physical phenomena. Negative probabilities give the possibility to extend the range of physical phenomena by considering phenomena which violate the principle of the statistical stabilization.
"Interpretations of probability." A. Khrennikov (1999)
A nice illustration of negative probabilities in the quantum world is described by Johannes Koelman in his blog post "Quantum Casino - Less Than Zero Chance".
Even with these limitations there is still the question of whether we can know anything (Scepticism), or even if we can know that (Pyrrhonian scepticism). However it is also often forgotten that the scientific method acknowledges that it's results may be incorrect and may revise what it has previously asserted upon observations of previously unobserved phenomena (hence adhering to Fallibilism).
Lastly there is always the possibility that there is actually no such thing as truth or that any assertion of truth is shorthand for reporting an infinite regress or even just syntactic sugar (Deflationism).
Therefore, it is my opinion that we cannot answer the question posed - it could be the case that superior knowledge acquisition methods are theoretically possible, it may even be the case that the scientific method is a stepping stone toward such methods, or it may be that what the culmination of human experience has produced is merely a reflective mask over the persona of reality, or even that the search itself is what makes reality manifest as it does for us.
Another couple of questions the scientific method cannot answer may enlighten us, "Does reality exist when unobserved?", and "Does reality abide by the same laws when unobserved as when observed?".