Some images get pixellated

I have a set of images of character avatars that I'm using for a 2D game. I cannot share them as I have signed an NDA to not distribute them with others. They are 639x639 PNG images and were made with a graphic tablet. They don't look pixellated at all but when I put them in the scene or play the game, they get pixellated for some reason. I've tried to reduce the resolution and played with the options in the Inspector for those sprites but I still can't get a crisp image. Other images work fine, and they've been made with the same technique.
I am using a circular mask of the same size as the avatars to make them circular but even if I disable the mask, the avatars are still displayed in low quality.

 

This thread helped me a lot. Especially this part:

Make sure this isn't a IRL-Physics/Reality problem. Can you shrink them using any other program or method without losing detail? Sometimes you simply can't shrink details without losing them. One can only display so many pixels in a low resolution & details are only automatically preserved so far. (ex. There is no way to shrink a 4K resolution photograph into a 160x100 pixel area without it being utterly decimated.)​