For instance, many people who are native English speakers have terrible accents (ie Texan, Scouse, Scottish, etc etc) that make it hard for other native English speakers to understand, even though they're nearly 100% "fluent English speakers". In fact this applies to many if not all languages.
As a curiosity, two of the most lyric works of literature ever written in the English language - Lolita and Heart of Darkness - were written by Vladmir Nabokov and Joseph Conrad, one Russian and the other a Polish immigrant to the US that had terrible foreign English accents.