Is it possible to change your ethnicity?

General Answer
Generally speaking, NO. Ethnicity generally refers to a category of people who identify with each other based on common ancestral, historical, social, cultural, culinary and national experiences. As a result, because it is impossible to change your ancestry, personal and familial history, and national history, it is nearly impossible to change your ethnicity.

Affiliation Changes
However, in cases where ethnicity is defined more by affiliation than by a particular appearance or physical attribute, a person can change their ethnic group.

The most common affiliation change that has ethnic implications is religious conversion in those cases where religious identity has become synonymous with ethnic identity. An example would be that Bosniak or Bosnian Muslim can "ethnically change" into a Croat or Bosnian Catholic, by converting to Catholicism. Or similarly if a Jew converts to any other religion (Christianity, Islam, Hinduism, etc.), he is seen to have forfeited both his Jewish religion and his ethnic claim to be a Jew. Note that in order for the affiliation to change, the person actually has to adopt a different religion and simply becoming Agnostic or Atheist is insufficient to be "ethnically converted".

Another affiliation change includes linguistic choices. This can be a way that ethnicity "changes" through the use of a different language and only works if language is the primary means of identifying between two otherwise similar ethnic groups. One example is the division between Aramean Christians or Assyrian Christians on one hand and Syrian Arab Christians on the other. The Arameans and Assyrians both speak different dialects of Neo-Aramaic, while Syrian Arab Christians speak Arabic. The language is one of the key ways that these groups differentiate between each other. If a Syrian Arab Christian were to move to an Aramean village and speak Neo-Aramaic, this person would "ethnically convert" to become an Aramean Christian. A similar case occurred during the Magyar Tribes' conquest of the Pannonian Basin (e.g. the formation of Hungary). Most of the people in the Basin were Slavs and, over the centuries, some married with the Magyar invaders, but most did not. Regardless, those Slavs who spoke Hungarian (the language of the Magyar) were considered to have "ethnically converted" to being Hungarians, while those Slavs who retained their languages, like the Yugoslavs, Czechs, and Slovaks, were considered to still be Slavs.

There is also nationalistic re-identification, which is simply that people will identify as a different ethnicity if the borders are redrawn. A perfect example of this is the development of the modern Macedonian ethnicity, which occurred in the 20th century when the government of Yugoslavia successfully controlled the westernmost portion of Bulgaria and proceeded to "de-Bulgarify" it. Now, most Macedonians no longer identify as ethnic Bulgarians, but ethnic Macedonians, even though, if you asked their ancestors 100 years ago how they identified, they would identify as Bulgarians.