Some countries just don't like the look of you and refuse you entry. I know of someone who got sent back from CPH or somewhere nearby, and everything the airline saw passport, visa etc. wise was spot on.
Human error, or erring on the side of caution perhaps such as the South African issue in another post.

Couple of examples mentioned by people I know at an airport:

Someone the other day (on another airline) got sent back from India as their visa had the passport number from their old passport on it and they only brought the new one. Check-in agent missed it.

Woman going to Jeddah for Umrah with her family. Visa was issued back in September, valid for a few months and all details were correct.
BP issued at check-in etc.
Got to the gate, and the agent noticed her passport was 1 week short of the requirement that it should be valid for 6 months from the date of entry (if she had gone in Sept when the visa was issued by the Saudi embassy, fine, but she went around 4 weeks later so the passport wouldn't have been valid at that point).

Possibly a word, number etc. was incorrect. Some countries are relaxed about say a passport number being 526789011 when it should be 52678901, or Muhammad instead of Muhammed. Others won't be.

At the end of the day, whoever is scanning the BP and checking the name and expiry date on the passport isn't going to be double checking visas, OCIs etc. They assume whoever was at check-in has done that and all is fine.