There are two different reasons that people might have a bad reaction to dairy products. Some high-fat dairy products (like butter) can be consumed by people who are lactose intolerant with minimal adverse effects.
Cow's Milk Protein Allergy
When people with cow's milk protein allergy consume any amount of protein from cow's milk (present in all dairy products) their bodies produce an immune response against the milk proteins. This reaction often occurs very soon after ingestion (seconds or minutes). Because the immune system response is probably not proportional to the amount of milk protein consumed, people with cow's milk protein allergy should avoid consuming any quantity of any kind of dairy product.
The prevalence of cow's milk protein allergy (CMPA) vary from 2% to 7.5%. (source)
When people with lactose intolerance consume foods that contain lactose (milk sugar), their digestive system is unable to break down the sugar into parts small enough to be metabolized. The milk sugars carry on further into the digestive tract and are fermented by bacteria, producing a large amount of gas. The amount of gas produced is proportional to the amount of lactose consumed -- a smaller amount of milk will produce a smaller effect.
Because lactose is milk sugar, any dairy product which is low in sugar also has a low amount of lactose. For example, butter is made by removing the buttermilk which contains most of the sugars, so people who are lactose intolerant won't have as much reaction to butter.
Liquid cow's milk always contains milk sugars, and therefore contain lactose. Even lactose-free milk still contains lactose, but has lactase enzyme added to assist with digestion of the milk sugars.