frostysprite (user #377394)'s comment on post #1969424 tipped me off to this. When uploading from Twitter, if an image is over a certain size, then https://pbs.twimg.com/media/blahblahblah.jpg:orig will be larger than https://pbs.twimg.com/media/blahblahblah.jpg:large. (For smaller images, the two are the same.)
I assume the bookmarklet handles this properly, but those who are uploading manually should be sure to change the URL to find the largest possible version.