I think you can call yourself near-native if you've lived in the country long enough that you forgot all the grammar and make all the same mistakes the natives make.

...after half a bottle of grape juice. After a whole bottle, you can call yourself a polyglot.

My best guess is....when I  will be near a native English speaker. 

I think it's when being a non-native speaker is hardly noticeable.

Maybe when native speakers mistake you for a native. But in most cases: never.