Published or widely known failures of Page Rank algorithm
yes this "problem" is quite well studied and google itself is researching ways to improve it. its called the word sense disambiguation problem and is under active/ongoing research, as its becoming a competitive feature between different search engines. quite simply, the same word has different meanings depending on context.

as for your more general question about "widely known failures of page rank", it is long known that it cannot necessarily distinguish spam web pages designed to fool it, and a related technique called a link farm with many external pages pointing to one site. in this sense google search is employing somewhat similar aims/techniques to spam-detecting algorithms employed largely/widely by mail providers.

google has proprietary/closely guarded techniques to detect/filter spam and these are a large part of the continually refining versions released. then advertisers attempt to respond with new techniques. this back-and-forth evolutionary dance/ "arms race" is also known as search engine optimization from the advertisers pov, but the term also somewhat applies to googles process of improvement! (SEO is a very large field with strong overlap with marketing and probably worldwide worth in the billions of dollars.)

somewhat similar, so-called "google bombing" where people create a mass movement to use words together to affect search engine results. it is somewhat subjective whether to call this a "failure" of the search engine or actually a "feature" (as from the perspective of those exploiting it, because in a way search engines actually by design measure meme prevalence in cyberspace!).

one method of dealing with word disambiguation is actually looking at trends in users personal search engine behaviors. in this way person A typing in a query will get different results than person B depending on their search histories.[1]

a related field called Latent semantic analysis helps with this somewhat by representing documents as sets-of-words with weights associated with each word, and the same word may have different weights in different documents or depending on document similarity measured by the vector cosine similarity technique.

an example paper from this field is K&amp;M ref below. it actually uses &amp; builds on results of existing google searches for word disambiguation.

yet another known issue is called "topic drift" where basically, if you search on two terms $x$ and $y$, if $y$ is more common it will tend to dominate and more results will be related to $y$ and "drown out" results with both $x$ and $y$. see eg [2]


Google, Bing Battle for Better Disambiguation, AI eweek
Google &amp; WordNet based Word Sense Disambiguation Klapaftis, Manandhar
[1] Google Now Personalizes Everyone’s Search Results searchengineland
[2] hits/pagerank &amp; topic drift tcs.se