Hickam's dictum is a counter to Occam's Razor coined by medical doctor John Hickam who noted:
"A man can have as many diseases as he damn well pleases."
This is simply a colorful way to point out that the application of Occam's Razor in diagnosing patients is iffy at best. It is simply not true -- or not consistently true -- that finding one diagnosis that fits all the symptoms is better than finding three. This dictum is often paraphrased in more formal circles to something along the lines of:
"A particular diagnosis should not be excluded solely because it does not appear to fit the principle of Occam's razor."
This was perhaps originally an off-hand comment, but has been taken up as a useful reminder in the medical field that Occam's Razor is often Just Plain Wrong. As it happens, biology is insanely complex, and if you look for the one obscure disease that explains all of a patient's symptoms perfectly, you may well be able to find it. But, to paraphrase Dr. Theodore Woodward, "When you hear hoofbeats, think of a number of horses, not a zebra."
However, this is not only a reminder that the likelihood of an ailment occurring needs to be considered in contrast to the likelihood of a person having more than one ailment, but also a reminder that humans are unpredictable in their sicknesses. It is entirely possible for a person to have only three symptoms of the flu, and all symptoms of yellow fever, but still be suffering from the flu... or the flu plus gallstones. Moreover, sicknesses tend to cluster; stress, old age, a weakened immune system, or any number of other factors can open the door to multiple diseases or injuries, which may in turn interfere with the body's ability to recover from each.