Probably not.
The thing to realize about European countries is that they're nothing at all like the U.S. In America, if you or your parents were born here, you're an American, no questions asked. In Europe (I'll use Germany as my main example since that's where I'm from and what I know most about) it's really all about blood. You can't just apply for German citizenship. You have to actually BE German, which means having another family member already have a passport or be a citizen of that country.
http://en.wikipedia.org/wiki/German_nationality_law
Keep in mind that even if you do discover your great-great-grandmother was born/lived there for a long time, that does NOT, by any means, make you German/European - whatever country you're thinking of getting a passport from. It's hard to trace back to an ancestor that you can prove it for. Germany, for one, has expanded opportunities to get citizenship, but it's not really for anyone to just go about and decide they want citizenship. For some people of Jewish descent, they have made it possible for them to become German cizitens again after having to flee from the Nazis, but it's nearly impossible to prove. For one, many Jews could have been born in Poland, moved to Germany, THEN fled. The Nazis burned pretty much all records Jewish citizens had, so proving their heritage is very hard, if not impossible.
PLUS, many or most European countries really dislike dual citizenship, so you would either be denied, or forced to give up your American citizenship first. If you do that and still aren't allowed a passport, then you're screwed, because you're rendered stateless, and have protection from NO government at all.
Having a great-great-great-whatever live in Europe isn't good enough to prove that you're of that blood. Most Americans can't prove it. Thus, Germany's RIGHTFUL citizens are protected. =) I hope it never changes.