Personally, I find everything that happens in Star Trek that doesn't happen in the real world, pretty incredible in general. Name anything and I'll show you why.
For me, there's still a difference between suspension of disbelief, and having trouble when stories become self-contradictory.
Take the transporter. Most likely such a device will never be built, given what we know of physics. But I'll take it under suspension of disbelief.
The transporter transporting Data's mother and not recognising she's not human, however, is one of those self-contradictory cases to me. OK, so we have a device that has to replicate a body/
structure down to the quantum level (which should leave absolutely no room for even the smallest deception, otherwise you cannot reconstitute the body), and yet registers her as a human ?