Knowing some native speaker/writers, I'd say it's certainly possible. What kind of writing are you doing? I can see pros and cons for literary versus genre fiction.
Literary fiction is, I think, much more dependent on the words used to get the story across. I think of it as long free-form poetry, a poetic prose. Perhaps others would disagree, but the language itself is important to the reader. As someone who would have to think carefully about word choices, I could see writing as a non-native speaker might be helpful because you'd have to be intentional. The phrasing differences between English and Whatever could either help or hinder the writer's work. It could either be fresh or confusing for the reader.
In genre fiction, however, the language of the writing is not quite as important. Obviously if something is too "off", the reader will pick up on it (I would note that this is the case for me even between British and U.S. English).
However, you might get into the idiomatic issue depending on your setting. I think you'd be safer placing your writing in a context that would be familiar to you, even if you're writing in a non-native language. So your story would take place in Russia if you're a Russian speaker, as opposed to setting it in pre-World War II Japan, for example.
And if you're writing in a first person perspective and your main character isn't a native speaker, then people might be a little more forgiving as readers. To join along with the struggles of another.