Discuss @David Flin 's article here
I'm less familiar with modern views on it, but a couple of decades ago, you couldn't move for all the times the myth did the rounds (usually in the format of: "I was out there for twenty years, and they still couldn't speak English when I left."
Grandad then claimed that he had thrown the exam on purpose because he thought going to Grammar School would make him less working-class. This, to a boy born in the late 60's, was mind-blowing; not only the sheer difficulty of getting a quality education for most children in the 1920's, but also that a relative would give up the only avenue open to better oneself in life because it breached class barriers.
Just because their accent is thick and tricky to understand,
For outsiders. I didn’t know that my parents had any sort of distinctive “Indian accent” until some of my friends and acquaintances talked about it when I was a kid.