There are a number of drawbacks to such a situation. It is expensive, and no one really knows how all the drugs in question might be interacting with each other. Patients are often not very happy about being treated in this way either.
Evidence based medicine clearly has a lot to commend it, but it seems to me that it has this major drawback, that it encourages caregivers to look at each condition a patient develops in isolation from the individual as a whole. Thus an individual who develops high blood pressure might be given a certain drug because the evidence is that it is effective in treating high blood pressure. Then they get another drug, because the evidence suggests that this drug is effective for gastrointestinal reflux, which they have developed. A bit later they are getting headaches, so they now take another drug which the evidence suggests is effective in treating headaches. And so, sometimes, it goes on, and the list gets longer.
In this scenario, each of the disease entities - high blood pressure, reflux, headache etc - are treated as if they are separate, discrete entities with no relation to each other or to the individual patient as a whole. This is clearly a questionable, but rarely questioned, assumption. Once the drugs start mounting up, this approach seems more and more inappropriate (although it is perhaps highly appropriate to the profits of the people who make all the drugs). Would it not sometimes be better to look at the patient differently? Rather than looking at a collection of discrete disease entities, look at the person in total. Look at them holistically. Try to see whether all these various problems are really manifestations of one underlying problem.
The way western medicine has developed seems to have made it not very easily able to do this. Its heritage is reductionist and mechanistic, and the power of the drug companies reinforces this bias. Which is perhaps why many people turn to alternative forms of treatment which try for a more holistic perspective. In the kind of medicine I practice, for instance, Traditional Chinese Medicine (TCM), all the symptoms in the above example could very easily point to one underlying disharmony. In fact in TCM we speak of illness in terms of ‘root and branch’ (ben and biao). Generally treatment needs to be targeted at the root of the condition, not just at the branches. If there is something wrong with the roots of a tree, you might only notice this by looking at the branches, but unless the problem with the roots is sorted out, whatever you do to the branches is just a stop gap.