In one scene, Max & Larabee are shown arriving at the bank, while the Chief monitors their progress from his office. In the next scene, the Chief & 99 are at the bank waiting for Max & Larabee to arrive.
Despite being a relatively common gimmick, being trapped in a vault is exceedingly unlikely. Vaults are carefully engineered to provide either an escape route and/or a long-term air supply for anyone locked inside. Even the world's most famous and impenetrable vault, the Bullion Depository at Fort Knox, has an escape system.
When the bank vault door is opened, it is clear there are no locking bolts built into the door itself, just two bolts attached to the outside. Instead of attempting to cut through the "twelve-inch thick iron-and-steel door", the handyman could have quickly cut through the relatively thin external bolts.
The bank officials are excited to meet Freddie the Forger because of the many (forged) checks his ex-wife receives from famous people. However, after the first such check was returned as a forgery, the bank would refuse to accept "famous" checks from Mrs. Freddie, and would want to prosecute Freddie, not shake his hand.
Max and 99 quibble over whether Daylight Saving Time will mean one more or one less hour of air for the Chief and Larabee. However, changing the time on a clock would not effect the volume of available air or the number of hours it could sustain life.
When Max & 99 arrive at Baffles' prison cell, they're told by his cellmate Freddie that they've just missed him. The lights then dim, implying that he has been executed in the electric chair. However, prisoners facing execution are housed on death row, away from the general prison population - if Baffles was set for execution, he wouldn't have a forger as a cellmate.
Although the show takes place in Washington DC, most exterior shots are done in LA. This is especially evident in the armored car sequence, where among other buildings, they pass the Orpheum Theater.