Denis, your story strikes a Romanian chord. The situation around here is even worse, from what I can tell. But it is quite a fascinating question, with different answers from different points of view.
For an economist, it is a matter of price formation. In the state system, Romanian doctors are paid a fixed (and miserable) wage, largely unrelated to quality or effort. The incentive to pocket bribes is huge, and patients know it so well. In the private sector (with transparent and varied prices for medical services), bribes are almost unheard of. Also, there is a more or less efficient market for bribes. Patients find out how much a doctor expects, usually from past patients, or from other doctors. Surgeons receive more than GPs, professors more than debutants, etc.
But I think there is something more about "medical envelopes", from a cognitive point of view. First of all, there is a vast asymmetry of competence between doctors and patients, which gives the former a large freedom of action. Is this pill better, or another one? Surgery or not? Home treatment or hospitalisation? To make things worse, the post-hoc reckoning is not very helpful, since most decisions may be medically justified, but you might also end up dead. The patient is at the mercy of the practitioner since she does not know what choices are better. The best way to make sure one gets the proper treatment is to insure the benevolence of the doctor, and a bribe is the simplest path to gain the doctor's amity.
Second, there is something special about this particular social exchange: the patient is dealing in an ultimate value - her health. Something everyone in Romania says is that there is no price too high to be healthy. (Paradoxically, giving up smoking somehow does not make the list - self-hint-hint-nudge-nudge). If people would risk not bribing a policeman to avoid a fine, they are extremely unlikely to jeopardise their health in this manner. One cannot afford to stick to abstract principles (like discouraging corruption) when her life is at stake.
Finally, there is something like a Maussian gift in the affair: one passes a fat envelope even without the explicit mention of an economic exchange. It is not that the surgeon would not operate without being bribed - the patient just shows gratitude without visible economic reckoning. Of course, under the veil of generosity stands the solid self-interest of the patient. The fat envelope is meant to make sure that no scalpel is lost in her belly. But no-one says it out loud. It's a "I know that you know that I know etc" which makes sure that the transaction is smooth and polite.
To end with a personal anecdote: I was (and to some extent I still am) very wary of giving out envelopes to doctors. A little bit of moral prudishness, a little bit of fear (what if he feels insulted?), a bit of monetary unsaviness. Those who are more competent in these matters reassured me: "just put the envelope on his desk - he knows what to do next" After all, he is the expert, and I am not.


I thought Dunbar's hypothesis was a lot softer than this. Something like:
"Social animals are observed to form structured groups, where the size of group and it's sub-units tend towards certain numbers, the average which is species dependent. This average group size is correlated to average neocortex size across a range of social species, which suggests that neocortex size be a limiting factor for group size."
One of the fascinating things about Dunbar's research is how rapidly people came to see it as deterministic and prescriptive; whereas reading Dunbar's original paper my sense was that his theory was descriptive, though with predictive power. It's as though there is another neocortex limit on the amount of time a theory exists before we treat it as an absolute; on on the amount of uncertainty we can tolerate at anyone one time.
I'd also note that Dunbar tells us nothing about non-social species, or about those social species with no neo-cortex (such as insects).
Dear Jayarava — as this post said clearly, the hypothesis bears on primate societies alone.. This quick introduction does not suggest it could apply to other species, much less to non-social species.