There two groups of deficiencies regarding the scientific method: deficiencies of the scientific method itself and deficiencies of use of the scientific method.
The first group (deficiencies of the method itself) results by that which is the very strength of science - the context of objective reality.
Here are some examples:
In quantum physics, where the assumption of objective reality and the observer/observed duality is shaken. To circumvent the problem, duality is pushed an iteration further: we observe the observer looking at the observed object.
In philosophy, the premise of objective reality poses a problem of coherency.
In psychology, it's problematic not to reduce humans to automata by insisting on objectivism.
The second group (deficiencies of using the method) results by misuse and abuse of those who use the scientific method:
Science is nothing else but a flexible religion (the assumption of axioms requires 'faith' of their validity).
Scientists who tend to neglect this become entrenched in their views, thus resulting in the 'scientific priesthood' aspect of the scientific community.
From the human attributes of greed and desire for personal acknowledgement we get the 'popularity contests' in science.
Like religions, science too can be used well for social manipulations (e.g. demand for submission).
Science is a set of tools and we decide what to do with it. Be aware of the flaws, and remember they are integral part of the method. Use the scientific method, for it is a good tool.
Remember the method is not infallible, nor are scientists demigods - but humans.