I’m no numismatist, but I do know a little about the political history of American currency, so let me expand a little on samclem’s post.
Before the Civil War, there was no standardization. Currency was issued by state-chartered banks, with no national regulation.
During the Civil War, two things happened: (1) the government started printing its own currency, or greenbacks, in order to pay its bills; and (2) the government restricted private bank note issue to nationally chartered banks, under federal regulation. I believe that part of that regulation was that the notes had to be a standard size. See here for some pictures of national bank notes.
After 1913 the government got into bank note issue as well, with the creation of the Federal Reserve. Private banks continued to issue bank notes until the Great Depression, and then were forced to stop.
So the size standardization appears to date back to the Civil War. The last size change, as noted, was in 1928-29. Why have we never gone to differently sized bills? I don’t know, but Americans seem to be awfully resistant to monetary changes. In my lifetime we’ve rejected the $2 bill, and the dollar coin twice, and we can’t get rid of the stinking penny. Must be something in the water.