I think, like most other psychological theories out there, there is a lot of truth to it; however there can be many exceptions. I think especially with today's youth, many Elementary school aged kids certainly do not obey adults simply because we are in a position of authority. Some kids have no sense of respecting an authority figure. That is certainly an issue that begins at home with the parents.
One problem I have with Kohlberg, whose model was emphasized at a couple of schools I've taught at, is that he seems to regard internalization of some norm or another as the pinnacle of moral development, and that it fails to account for the real world, where dilemmas are more complex (and, I think, not necessarily solely defined by Western ideals of justice) than Kohlberg allows. With respect to say, gender, what Kohlberg is claiming could relate more to internalization of gender roles, or, for that matter, culturally defined behaviors, than moral development. In other words, what is guiding decision-making at the sixth stage of development is still defined by external factors, I think.
One of the things that struck me about this theory when I first read it is how he sees abiding by convention as a lower level of moral development. I grew up in a culture where people are expected to conform to traditional social rules. Order is valued over individuality and getting along with others is prized. Kohlberg's ideas, then, seem to privilege Western values and to say that they are morally superior to those of some other cultures. I think, therefore, that his theory is somewhat West-centric and not really objective.