Elk (an American Indian) was denied US citizenship because he owed immediate allegiance to his tribes, and they were not considered part of the people of the United States at that time.
https://en.wikipedia.org/wiki/Elk_v._Wilkins He was deemed not meeting the "subject to the jurisdiction" of the US part of Section 1 of the 14th Amendment.
http://www.cis.org/birthright-citizenship This analysis seems to indicate that the 14th Amendment does not grant automatic citizenship to babies born in the US and seems to interpret the "subject to the jurisdiction" as pertaining to the parent's foreign country - not as my plain reading of the section seems to indicate - to the baby.
Congress is supposedly working on this issue - in regards to changing the automatic issuance of citizenship to children born in the US of foreign parents.