Why didn't the Allies keep Germany from re-arming after World War 1?

They were certainly naive about that one. The idea was that economic sanctions should have made it difficult enough to re-arm. Then some of the more liberal European nations didn't really want to follow through on requirements that Germany not re-arm. The result was, nobody was inspecting Germany and the sanctions coupled with GLOBAL the economic crash of the 30's resulted in a very disgruntled German population. Hitler made promises, blamed the German problems on everybody else then proceeded to fund his war on the graves of Jews, Gypsies, homosexuals and anyone else that he determined to be "inferior". After WW2, the rest of the world was so determined to keep Germany from re-arming that they made certain that the nation was split into 3 parts. It wasn't until relatively recently that Germany was re-united.