Things get confusing (for many people) when numbers start to get bigger: The 20th century were the years 19xx (mostly). Did the 2 millenium end on New Year's Eve 1999 or 2000?
The latter, logically speaking. The mass media managed to convince the ignorant masses otherwise, but there you go.
It is ironic in this context to note that people would have got this right intuitively if years, and centuries, had been zero-based rather than the supposedly "natural" one-based system we have. If we'd used a zero-based system, with century 0 starting in the year 0, then the year 2000 would have represented the start of century 20, rather than the end of the 20th century. Problem solved.
Instead we have a silly situation where technically centuries are 1-based, but people constantly try to press for a 0-based system that seems more natural to them. People want the 21st century to begin with the year 00, not the year 01!