There is a widespread perception in some quarters that "native English speakers don't speak their language properly".

Here on italki, I've occasionally seen exchanges between members writing in their L1 who make comments along the lines of "Don't listen to those native speakers - they'll tell you any old nonsense".

There is one non-native teacher of English in particular who makes a point of telling me that my grammar is wrong, commenting on my answers with 'Sorry, but....'. She once told me - without a hint of irony - that I should come to her for lessons in the proper grammar which she teaches, rather than confusing students with my 'regional variant'.

Yes, these notorious "grammar-nazis" are so irritating...

However, I have never seen any foreigner who really has enough knowledge to correct my grammar in my native language (Russian). 

Perhaps it's an English-specific problem. English is an international language and has immense number of learning materials with explanation of very subtle nuances (sometimes too subtle). So, proficient non-natives sometimes really know grammar better than some native speakers. But in my opinion your story is about arrogance of some people rather than about understanding of grammar itself. Do they sincerely think that they know language better than native speakers if they are able to spot a subtle and unimportant mistake? I don't agree with them.

Yes, I have had non-native speakers attempt to correct me, though it was not as extreme a situation as yours. It does not happen very often but they were wrong each time. In one situation, a disgruntled student who didn't understand the difference between "a" (a verb form) and "à" (a preposition) was attempting to show me up in front of the class because he felt I had been very unkind in giving him the poor grades his work deserved. All he ended up revealing was his own ignorance of basic (really, really basic) French grammar, despite being in an intermediate class. That was funny.

Some non-native speakers who reach a decent level become eager to show off and catch errors in their target language everywhere. Unfortunately, though they sometimes give useful feedback for very obvious errors, these people also tend not to have a thorough enough grasp on the language to identify fluent native speech and writing. Instead, it seems strange to them and they assume it is incorrect. They tend to be stuck in the framework of their native language and have difficulty accepting that another language defines the logical and possible differently.

That said, I have nothing against non-native speakers correcting native speakers when necessary. As long as one knows what one can correct and the limits of one's knowledge, it's fine.

The real problem is people, native or not, with incomplete knowledge, making corrections when none are needed. I've seen native speakers "correct" non-native speakers, and in the process reveal how little they knew of their own language. Assuming one knows best just because it's one's native language is a form of arrogance as well. If one hasn't studied and read a lot, there is a lot one doesn't know and cannot correct.

You are right, there are characters who sorely tempt us to bring our calcaneus in contact with their gluteus maximus. Hubris never served anyone well. On the other hand, one must learn something for many years simply to know how much one doesn't know and never will. Fools rush in ...

As Nikita said: "But in my opinion your story is about arrogance of some people rather than about understanding of grammar itself." - There's nothing to add!