Whether they were purposefully accused or wrongly.
There are miscarriages of justice across the US today in which people who are falsely found guilty are executed each year, and with a genocide the scale of the Holocaust, of course there were people who claimed not to be Jewish but were still liquidated.
The common answer for how Jews were selected is that you were Jewish if your grandmother was Jewish (observant, religious and openly so). And of course, it is true that they used this as one of their methods for selection. Because of such a blunt system of selection, there were many times people argued that they had never practiced Judaism, observed it or were even religious, but these people would have nearly always been ignored. There were other methods the Nazis used for selection too which they believed determined if someone was Jewish. The Nazis believe that Judaism and "the Jewish problem" was more than just a region, they often used the term "Jewry" to describe the "untermensch" they killed. To them, Jewry was a cultural concept as well as a spiritual concept, and so they also selected Jews through census records that dated as far back as 1901, where Jews believed they had nothing to fear by openly declaring their religion and culture. They also used tax return records, and synagogue membership lists.
Perhaps more horrifying is the fact that they used parish membership lists to find converted Jews. The largest number of "Falsely convicted" Jews came from this quota. They believed that Jewry was something that you could not escape, and that it was a biological issue as well as a cultural and spiritual one.
To answer your last question, I'm sorry to say it's almost impossible to know the numbers. There was not one Nazi held tribunal or court session to determine if someone was Jewish, and so there are practically no records of people denying their Jewry except from the odd journal.
fyi, you'll find some previous responses in this thread
To piggyback on this...
Why didn't more people just claim they weren't Jews, move to a new town and live like everyone else?