People who think they’re Jewish but are not include those
1. Who converted via non-Orthodox
2. Who have a Jewish father and non-Jewish mother (paternal descent)
3. Who were adopted by non-religious Jews
4. With a supposedly Jewish mother (and father) but the mother’s mother (or mother’s mother’s mother, etc.) is one of the above (#1, #2 or #3) or maternally descended thereof any number of generations back.
And such people (#4) could even honestly mistakenly think they come from a fully Jewish heritage, on all sides. And will claim as such if asked (for kiruv or marriage purposes.)
And #4 can go back quite a few generations (i.e. five+ generations back) considering that the Reform movement is 200 years old and in the United States intermarriages have been a consistent issue for the last 150 years and false conversions even longer (as those started in 1800s Germany.)
In short, a large percentage of self-identified Jews who don’t come from a straight line of Orthodox descent are, in fact, gentiles.
And identifying who is not Jewish based on maternal descent over the past 200 years is an extremely difficult task in many or most cases, requiring a very large amount of research and investigations that in 2019 may or may not result in a definitive determination.