Most of my linguistics texts back that up. Takes three generations for immigrants to assimilate linguistically (from a linguistic view, so cultural stuff is separate). The first speaks only their native language and a bit of the "new" language. The second generation grows up bi-lingual, and the third only learns a tad of the native tongue and most of the new language. Generalizations, but they often hold true.
Hmmm,
My parents emigrated from Sri Lanka to Australia as borderline bilinguals who struggled to understand what was being said to them when they arrived. Now, however, both of my parents speak English almost as frequently their native tongue if not more. Home conversations nowadays are split almost 50/50 Sinhalese to English.
I was (kind of) bilingual during my earlier childhood and originally spoke Sinhalese far more frequently (and probably more competently) than English, though I can only write in English. As I got older and grew up in an English-speaking educational system, that reversed to the point where I can top the first semester English exam for the year and understand about 50% of Sinhalese in general (about 95% of what my parents say to me) and lack the ability to actually speak the language in a sensible manner (with my accent making it hard to pronounce several of the more common sounds).
My younger sister on the other hand fits into what looks to be by that statement, someone who's a third generation immigrant. She's grown up speaking English and learned how to read and write in it after I taught her some of the basics prior to her entering school (much to the surprise of her teachers and classmates

). In contrast to this, she doesn't recognise more than about 50-60 words of Sinhalese and doesn't speak the language at all.
I'd say my parents fit more into what you'd call the second generation of immigrants, however.