Poland was occupied by Germany during WWII. The people still considered themselves Poles during this time, and the only reason there wasn't a 'Poland' was because of military strength, so it seems silly to say that early 1939 Poland and late 1944 Poland are different countries because of lack of continuity. Certainly, almost all Poles will tell you the country 'began' (it kind of had a soft start) in 966, and not in 1944.
(Note: there was technically the government-in-exile in London, which you could argue maintained continuity, but I don't think it's necessary so I'm leaving that out of this.)
This (and most other examples here) could just as easily be explained by nationalism. specifically nationalist projects to claim a much older heritage for the current nation state, to legitimize the current nation, and to give it a glorious and hard fought past so people will be proud of their country.
Personally I think a lot of the idea of countries having existed 'for a long time' is the result of these nationalist projects that all occured in the 1800 hundreds
(Note: there was technically the government-in-exile in London, which you could argue maintained continuity, but I don't think it's necessary so I'm leaving that out of this.)