I think some historical perspective on this might be useful. If you look at Choki Motobu, one of the icons of classic, traditional Okinawan karate and a pioneer in the movement to bring karate to Japan, you'll find that he had studied with no fewer than five masters of the previous generation. And I suspect that the same would prove true for several others of the more revered figures in Okinawan karate.
I've wondered for a long time if the whole quasi-sacred character of the MA student/teacher that's enshrined in popular culture and our own thinking about the MAs isn't something that came in via the transformation of Okinawan karate into a Japanese MA in the militaristic prewar era. We know that Funakoshi strongly believed in the military utility of karate, not as combat training, but as a group activity to instill reflexive obedience into young conscripts for what everyone over there could clearly see was the coming Asian war. And that kind of semireligious loyalty was a big part of post-Meiji Japanese culture, with the Emperor identified, literally, as a god. My own feeling is that the Okinawans were far more pragmatic about it; their own tuite styles were probably family-rooted, like the seemingly endless variants of the CMAs, and it would make sense to follow a single teacher there, because you learned te from the most experienced, knowledgable practitioner in your own family—no one else was going to teach you their family secrets, after all! But Motobu, who for family reasons had to pick up his training where he could get it, bounced around quite a bit, and that may well have been the pattern for those in Okinawa who were not the 'designated inheritors' of the family style. The Japanese ethic of total obedience and unquestioned loyalty may turn out to have been the source of this 'selfless devotion' quality that's supposed to characterize the MA student in much popular writing and dojo/dojang folklore. That doesn't mean that individuals may not, for their own reasons, feel personal loyalty to their instructors, just as they may to a particular bookstore that they patronize because it's a small, locally-owned enterprise run by people who really love books and know a lot about them, and whom their customers don't mind paying much higher prices to than Amazon charges, in order to support them. But as jks pointed out earlier... that's an individual choice, based on personal preferences and values. It doesn't inherently come with the territory.
I heartily agree with what people in the previous posts have been saying about the American free-market/service economy ethic (knowledge is a commodity like any other, you pay for it as part of a straight exchange and that's that) making a poor fit with the personal loyalty ethic that the MAs have come to embody. But I do wonder if, with a bit more probing, we might find that when you remove the practical bases for single teacher training in much earlier times—namely, the family-based nature of MA knowledge, the relatively low numbers of accomplished practitioners, etc—you find a good deal of dojo-hopping back then, as well.
*) or that they can dodge bullets or they can beat up gangsters or what have you. I am sure many of us have heard "Oh so you are a Black Belt. What are you going to do if I do, ... "
