Hannover vs Baden

After immigrants were in the US awhile, it apparently became the fashion for them to name the nearest "big city" as their places of origin on censuses and certificates ---- especially among those living in large US cities who didn't want to look like they had been "country folks" or "farm hicks."

In my family, the Hannover became "Bremen" (which I think was actually an independent city-state and was never a part of the Kingdom of Hannover --- is that correct?), and the Prussia became "Koln" after 1880.

But I have no real way to know if either of those ancestors actually lived in those cities.

Does anyone else have some experience of this phenomenon to share with us?