From this article: http://www.slate.com/articles/news_...2004/08/why_puerto_rico_has_its_own_team.html
And this one: https://www.si.com/olympics/2016/08/06/olympics-puerto-rico-united-states-ioc
OT, but frankly, I think the whole 'being American citizens, yet not being able to vote for the President or the House/Senate' thing is really weird.
It is because of the way the Constitution is written.
Technically, citizens don't vote for President. Individual electors, who represent the will of voters in each state, vote for President. There is no provision for electors from non-states, other than DC, to have a voice in the race for President.
Similarly, the Senate and House of Representatives are elected by States. Not territories or possessions. The House of Representatives does include non-voting members from non-States.