I think most immigrants I know have an accent when they speak my language, even if they have good grammar and vocabulary, and have been living here a long time.
My dad has been in the US far longer than his birth country of Japan, having moved in his 20s. His English is excellent in terms of vocabulary and grammar, but his accent is very strong. I, on the other hand, having been exposed to the language when young and living in Japan for just one year, have a much better Japanese accent than he does English but am far worse in all other aspects.