I have the TPS55340EVM-147 board that I am testing with. The only thing I am changing now is the inductor. Originally I did not have an inductor that met the calculated criteria for Isat. I used a Coilcraft MSD7342-472 which has an Isat of 4.1A for 10% drop. My calculation showed that it should be 8A.
I went ahead and got the same type of inductor as was originally installed on the board, just in a different inductance value. I got a Coilcraft MSD1206-472.
Originally using the 7342 I was able to get over 670mA of output (3.3V input and 12V output). However, when I replaced it with the 1206 I wasn't even able to get 470mA. I fully expected the new inductor to make the part work better, but it in fact made it worse. I looked at the specifications for both parts and I do not understand why this would happen. I can go ahead and use the 7342 since it seems to work, but I need to understand why this occurred so I can create a reliable design.