Other mistake: When the radio operator opens the Aircraft Communications Codes book, he turns first to a 'L' page, but all the codes are 'F' codes including the key code. He then turns to a 'J' page and again the codes are the same 'F' codes. He then turns three 'E' pages, pages one and three are identical.
Plot hole: The attack on Burpelson should have started at night. Using time cues like Turgidson's secretary stating the message was "about 8 minutes ago" and "it's three o'clock in the morning" (the clock says 3:15) Mandrake sent the go code approx 2:06am (CST). Later in the war room Turgidson states the code was sent "about 35 minutes ago". Ten minutes later the president orders General Faceman to assault the base. This order would have gone to Alvarado approx 2:51am, about 30 minutes to load troops and drive seven miles and they should have reached the base perimeter by 3:20am (with headlights).
Plot hole: Turgidson's 'secretary' listens to the incoming telephone messages for only a second or two, and relays a much longer message than she could have heard during that time.
Answer: Peter Sellers improvised it. During that scene if you watch the actor playing the Russian character you can see him trying not to laugh at Seller's performance.