ETA: Of course, Puerto Rico is 'the US' so this doesn't apply in this case, but interesting nonetheless.
Dual citizenship rules differ from country to country. For example, a friend of mine was adopting 2 children from Russia. When she went to pick them up, they were told that the boy's mother was from Kazakhstan and even though she had given birth there and given up the baby there, he was not a Russian and the 'Baby House' could not adopt him out. It was very sad.
Here is an article I found about it dual citizenship. It's long, but interesting:
Brief overview on dual citizenship
In general, countries define citizenship based on one's descent, place of birth, marriage, and/or naturalization. That is, you might be a citizen of a given country for one or more of the following reasons:
You were born on territory belonging to, or claimed by, that country (often called ius soli, or sometimes jus soli -- Latin for "right of the soil").
One or both of your parents were citizens of that country (often called ius sanguinis or jus sanguinis -- Latin for "right of the blood").
You married a citizen of that country (though please note that the practice of granting immediate, automatic citizenship to a foreign spouse is far less prevalent today than it was decades ago).
You (or one or both of your parents) obtained that country's citizenship by going through a legal process of naturalization.
The exact details will, not surprisingly, depend on the laws of the country in question. For example, the US limits its application of ius sanguinis by requiring American parents to have lived for a certain period of time in the US before foreign-born children can be entitled to US citizenship by birth. Many countries (Switzerland is one example) do not confer citizenship via ius soli at all, and those which do generally make exceptions for children of foreign diplomats. Automatic citizenship via marriage is rare nowadays; more commonly, marriage may allow one spouse a "fast track" to immigration to the other spouse's country, but a period of non-citizen permanent residence would still be required before the immigrant spouse could obtain a new citizenship via naturalization.
Since there can be several ways to acquire a given country's citizenship, it is possible for someone to be considered a citizen under the laws of two (or more) countries at the same time. This is what is meant by dual (or multiple) citizenship.
For example, my son has been a dual citizen of both the US and Canada from the day he was born. He is a citizen of the US (via ius sanguinis), because his parents are both US citizens who fulfilled the US's legal requirement of residency in the US prior to his birth. And he is also a citizen of Canada (via ius soli), because he was born in Canada and neither my wife nor myself were in Canada as foreign diplomats.
I, too, am a dual citizen of both the US and Canada -- a citizen of the US because I was born in the US, and a citizen of Canada because I went through the Canadian naturalization process (an action which did not cause me to lose my US citizenship).
Countries usually frame their citizenship laws with little or no regard for the citizenship laws of other countries. In my son's case, for instance, the US does not care that Canada thinks he is a Canadian citizen, and Canada does not care that the US thinks he is a US citizen.
In some (but, please note, not all) cases, a country may seek to restrict dual citizenship by requiring one of its citizens born with some other citizenship to renounce (give up) the other citizenship upon reaching adulthood. Similarly, newly naturalized citizens in some (but not all) countries are required to renounce their previous citizenship(s); the US has such a requirement, for example, but Canada does not. And in some (but, again, not all) cases, a country will automatically revoke the citizenship of one of its citizens who acquires another country's citizenship by naturalization, even if no explicit renunciation was involved.
Where one country requires a citizen to renounce the citizenship of another country, this renunciation may or may not be acknowledged or accepted by the other country. This can sometimes lead to sticky legal situations. Also, countries which require such renunciations differ in how seriously they treat this requirement. In some cases (such as Singapore), an applicant for naturalization may be required by his new country to go to an embassy or consulate of his old country and renounce his old citizenship in a manner prescribed by his old country's laws. Other countries (such as the US in recent years) may treat their own naturalization oaths' renunciatory language as essentially meaningless and take no steps to enforce it at all.
As a general rule, dual citizens are not entitled to any sort of special treatment by their two countries of citizenship. Each country will usually consider the person as if he were a citizen of that country alone. Some people describe this sort of situation by saying that a given country "does not recognize dual citizenship" -- but this usage can be confusing, because it might mean either that a country passively ignores other countries' claims on its citizens, or that it actively prohibits its citizens from also being citizens of other countries.
Citizenship frequently carries with it legal obligations relating to taxes, military service, and/or travel restrictions. Again, since countries usually insist on dealing with their citizens without regard to any other citizenships they might hold, and tend to frame their laws regarding citizenship obligations without regard for the laws of other countries, a dual citizen could possibly find that a country which considers him a citizen, but in which he does not live, expects him to pay taxes (possibly in addition to taxes he is already paying in his country of residence); considers him liable to be drafted into its army (even if he has already served or is currently serving in the other country's army); and may forbid him to travel to certain countries, including possibly his other country of citizenship.
In practice, such situations are often smoothed over via tax treaties and the like, but conflicts could (and sometimes do) occur. Also, be aware that most countries (the US is the main exception) base liability for income tax on residence (where one lives) and/or source of income, not solely on citizenship; thus, dual citizenship usually does not automatically translate into double taxation.
Citizenship claims by a country over a given individual could happen even if the person in question never sought recognition as a citizen of that country -- or even if the person was totally unaware that he/she was a citizen of that country according to its laws. Accordingly, anyone who is planning to travel to an ancestral homeland -- even for a brief vacation trip -- would be strongly advised to check that country's citizenship laws carefully beforehand. Otherwise, the trip could run into unpleasant snags if you discover, say, that Country X considers you to be one of its citizens because your father (or even your grandfather!) came from Country X -- and that, as a result, you need a passport issued by Country X in order to leave -- and in any event you can't leave until you have put in a year's worth of military service in Country X's army -- and when consular officials of the only country where you thought you were a citizen try to intervene in your behalf, they are told to get lost because your case is strictly an internal matter between Country X and one of its own citizens (i.e., you)!
On the other hand, dual citizenship can have distinct advantages. In particular, a person with dual citizenship has greater flexibility in his or her choice of where to live and work. Thus, it behooves anyone with dual or multiple citizenship -- or with the possibility of claiming such a status -- to investigate the pros and cons of the specific situation very carefully.
http://www.richw.org/dualcit/
. I guess I wonder in what capacity exactly wikipedia or whoever is saying he's puerto rican. In terms of citizenship, no. In terms of cultural identity or national identity, no. In terms of some other weird definition of "puerto rican-ness", maybe 
