the results when using the at parameter for different names is very inconsistent.

For example, ?at=0 can be used to get the first name, but this only seems to be working for accounts that have been renamed at least once:
https://api.mojang.com/users/profiles/minecraft/Djinnibone?at=0

or accounts that are legacy:
https://api.mojang.com/users/profiles/minecraft/noobkackboon?at=0

however, it does not work with (non-legacy) accounts that haven't been renamed:
https://api.mojang.com/users/profiles/minecraft/Notch?at=0

I am not actually using this to find the first name of a user but it's an inconsistency in the API. You can replace 0 with any timestamp before the account's existence (the date may be unknown so such a request isn't too uncommon).

I guess this is related to different handling of usernames/dates since the name changes have been introduced.
That also leads to the question what the API returns for the same query for a user created after the name change. Are there any differences?