The conventional wisdom in medicine is almost the opposite: more pills leads to lower compliance. This has been studied and will be studied more the future, since it is an easy source of patents. The term to search for is "fixed-dose combination," putting two drugs in one pill. Really, how could it be otherwise? It could: adding an "important" drug to a regime of "unimportant" drugs could increase compliance of the "unimportant" drugs. But I think this is most plausible with vitamins. Being in a study might cause that particular drug to get labeled "important," but I doubt it.
As I said elsewhere on the thread, it is well-known that medical studies increase compliance. The conventional wisdom is a more specific hypothesis that proximity in time to an appointment increases compliance with all drugs, a claim that has consequences outside of clinical trials. I was not able to find any studies of this, perhaps because I don't know the correct terminology. Compliance is a booming field, especially outside of clinical trials, and I expect that this will be well-studied in the future if not yet.
In other words, it is important to distinguish between the placebo effect and the effect of being in the placebo arm of a clinical trial. The pure placebo effect of placebo vs no drug is very rarely studied.
The placebo effect (benefit in groups receiving fake pills) and nocebo effect (detriment in those same groups) have frequently been the bane of medical research. They are usually explained in terms of psychology: because people receiving placebos believe they have been treated, they get psychosomatic effects that cure symptoms and create side effects. This explanation is supported by the fact that the placebo effect is strongest when the effect being studied is subjective - eg, tests of painkillers and antidepressants. This explanation is neat, tidy, and in my opinion, altogether unsatisfying.
I have an alternative theory. Most people in medical studies take more than one medication; in addition to the drug being studied, they take unrelated drugs and supplements, usually including a multivitamin and often including other things they were prescribed. However, many people take their pills inconsistently; they miss or mistime some fraction of their doses. This is especially true of depressed people. Prescribing a placebo, however, fixes this; when they take their placebo pill in the morning, they are reminded to take everything else they should be taking. In addition to making pill-taking more salient, being prescribed a placebo may also cause some people to fix the organization and affordances they have for taking pills.
I suspect that many of the benefits attributed from placebos may in fact be due to increased compliance with unrelated prescriptions and correction of vitamin and mineral deficiencies. Arranging a study to test this should be fairly straightforward; simply measure the rate at which unrelated prescriptions are refilled in two groups, one of which receives sugar pills and one of which does not.