The experiments were performed in a proud university town. What Milgram actually demonstrated is that those people trust scientists and believe in the value of science. In other words, they obeyed commands when those commands seemed to align with their preexisting worldview and ideology.
The inspiration of these experiments was Adolf Eichmann's plea that he was merely following orders. Milgram obstensibly set out to disprove that, but is said to have inadvertently proved Eichmann's plea plausible; if Milgram showed that people do follow orders, then maybe Eichmann's innocence plea was plausible. But we know Eichmann was not simply following orders. There is ample evidence that he was enthusiastic about the Holocaust and went above and beyond what was required of him. Eichmann may have been following orders, but they were orders he was ideologically aligned with. Just like the Milgram subjects were ideologically aligned with the value of science.
Many variations on the Milgram experiment were conducted. When the Experimenter didn't dress like a scientist, compliance was lower. I didn't hear this until years after I was told about the experiments in my undergrad psychology class. When I learned this, I felt deceived. It turns the mainstream conclusion on it's head.
If we were to run a modern experiment more applicable to Eichmann's 'following orders' plea, the Experimenter would be dressed like a cop and the Learner would be a black man. How many people (except the overtly racist) would follow orders in this scenario, when the orders were so flagrantly in violation of their personal values? Virtually none I assert. That's the obvious water is wet answer, while Milgram's answer is the shockingly unintuitive headline grabbing conclusion. Milgram's conclusion asks us to believe there is an Adolf Eichmann lurking inside all of us, waiting for heinous orders to execute. That just doesn't jive with what we actually know about Nazis.