TLDR It depends. Evidence suggests that many cats at least, can drink milk in some forms, particularly unpasteurized milk.
There was a famous series of studies done on, among other things, cats and milk consumption, by Dr. Francis Pottinger, in the 1930s and 1940s.
As the study has been around so long, the results have been analyzed time and again, and you can find much on google.
Wikipedia summarizes the findings of the study quite well:
In ... the "Milk Study," the cats were fed 2/3 milk and 1/3 meat. All groups were fed raw meat with different groups getting raw, pasteurized, evaporated, sweetened condensed or raw metabolized vitamin D milk. The cats on raw milk were the healthiest while the rest exhibited varying degrees of health problems.
From this we can see that some forms of milk are indeed bad for cats.
I am searching for the original text of the study to cite here, which I would expect to be public domain by now, and so far have not found it.
There are a number of attributed differences between pasteurized and unpasteurized milk, which may affect the health of a cat
, but the simplest to recognize is that of lactase (note the distinction from lactose: Lactose is a sugar, lactase is the enzyme which helps digest lactose). This is the simplest to recognize, due to the nearly immediate symptoms associated with its absence, as well as the well understood mechanism by which it works.
The main culprit is milk’s lactose, which many cats have trouble digesting. The result: diarrhea or stomach upset. Not exactly romantic.
Lactase is an enzyme naturally present in cow's milk, which is essential for the complete digestion of lactose (milk sugar). The process of milk pasteurization destroys this enzyme, thus leaving the consumer of the milk (a cat in this case) to his own devices to digest the lactose, and many cats (and other animals) don't have the gut flora necessary to digest it, thus leading to digestive problems.
The above part of my answer appears to be incorrect (thanks to @Scott for raising the question); although that doesn't change the results of the above quoted study; only the mechanism by which these results can be explained. I will try to edit this question as I find additional information.