In addition to the events in 1991 mentioned by reximhotep, as recently as 2009 an article was published that claims to have confirmed the Romanov children were identified using DNA evidence. This all but puts to rest the claims that Anastasia Romanov made it out of Russia in 1917 or 18 as far as I am concerned, and fits with the historical evidence we have that they were executed. It seems that the whole family has been, by now, accounted for.
http://journals.plos.org/plosone/article?id=10.1371/journal.pone.0004838
The story itself is not otherwise hard to believe. I don't know who she was, but a Russian woman named Anastasia, possibly of noble origins, escapes Russia during the Revolution or Civil War and ends up living out her life in the Philippines. But the evidence presented in the article you've linked, that she was actually Anastasia Romanova is pretty suspect. There is a photo that kind of maybe looks like a painting, a name (and let me tell you, you can not walk 5 feet in Russia without finding two people with the same name), a mention of jeweled walls and fresh bread. That does not to my mind a convincing argument make.
To the best of my knowledge Anastasia's body was found 1991 in Ekaterinenburg with the rest of her family and was subsequently buried in St. Petersburg in 1998. DNS-tests have shown the identity of the imperial family. A great source for the fate of the doomed Czar's family is "Ekaterinburg: The Last Days of the Romanovs" by Helen Rappaport.