I found a significant error in a paper with 1000 citations. My advisor agreed that it is a significant error, but discouraged me from telling the author.
It occurred to me that maybe other graduate students also found the error, and never told the author, or they did tell the author and he just never corrected it. I think an author would have very little incentive to correct his own work, especially if they were famous for that work, and readers are strongly incentivized against writing a correction paper, because the author would be mad at them.
How often does this sort of thing happen? Can we really trust the literature?
I'm not sure what you are looking for. I know for sure that the seminal paper that made one of my bosses famous had glaring mistakes in it. Despite the fact that his result was completely wrong, the paper started a new research field and is now highly cited (around 1700, I think). No one ever bothered to point out the calculation mistakes, without which the paper would have never gotten accepted, because his proposal got confirmed experimentally. More detailed calculations done by others showed his mistake, but confirmed his intuition. In any case, I can't find survey data in my field dealing with un(der)reported mistakes in papers. But, as you do research, you are bound to find quite a few.
On the other hand, I found this oncology paper on unreported mistakes in oncology papers http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3031354/
Looks like less than 25% of those who spotted mistakes in the papers, actually went on to report them.
The practice in my field is to write the authors about the possible mistake. If they are willing to correct it, there is not much point in escalating. If not, you can write a comment on how wrong the paper was, post it on arxiv and send it to the journal editor. This, assuming you're sure they made a mistake. Most people aren't willing to go through all this pain, but I've seen this many times.