I'm sometimes half tempted to do this for a bit of fun...
We have a running joke in my family that my sister and I aren't really related.
My blood group is B+. My sister is O+. My mum was O+. (I'm 100% sure about that because when I was 18 I got into blood donation by going along with her. But it's not critical to the story). And my dad was AB+. But that's where it starts to get awkward. My blood type is perfectly compatible with my parents': my genotype is BO with a B from my father and an O from my mother. But my sister's isn't: with parents who are AB and O you can be A (genotype AO) or B (genotype BO, like me), but you can't be O. Since my sister is O, it follows that her parents can be O, A (genotype AO), or B (genotype BO) - in fact, anything but AB.
Obviously there was a mix-up at the hospital, we joke. Except that doesn't work, because I was born at hospital and she was born at home, so if there as a mix-up it was with me, but I'm not the one whose parentage is in doubt.
Maybe my dad wasn't AB. Maybe he was B (genotype BO), which would explain things. But that seems highly unlikely. He died a long while ago so unfortunately we can't ask him. But my mum was always positive that he was AB - it was typed when he did his national service in the RAF, and he was a blood donor - and his sister (my aunt) says he was AB. Before she retired she was a consultant haematologist, so you'd think she ought not to get that wrong.
So the obvious next part of the joke is that my sister had a different father. That's no big deal these days, of course, but it was in the early 1960s! However that doesn't work either, because she looks like "my" dad and I don't, and anyway she's only 11 months younger than me, and neither of us can imagine our mum cheating on my dad at any time, let alone when she would have had a 2-month-old baby!
So it's a great mystery. It would be interesting to settle it. But perhaps it would be more fun in the long run not to....