Tomorrow is Yesterday - S1-E20
Stupidity: The Enterprise is very concerned about John Christopher going back to earth and changing history, but at the same time he has free access to go anywhere in the ship, and all his questions are being answered.
Stupidity: When Gary 7 learns that agents 201 and 347 were killed in an automobile accident, he must take over the mission and save the earth from nuclear war. However, their technology allows them to transport anywhere they want through the vault device. It makes no sense for both agents to drive to McKinley Rocket Base when they could have easily transported there.
Suggested correction: Good point, but it seems more likely that 201 and 347 were executing a careful plan that depended on their cover as normal humans acting normally. Using transporter technology risks discovery. Gary Seven only used the vault transporter as an emergency measure, missed being observed materializing by mere seconds, and was detected by a guard moments later. Same for Kirk and Spock.
Stupidity: Why, oh why, didn't Kirk bring a communicator with him when he beamed down and found himself on the fake Enterprise?
The Enterprise Incident - S3-E2
Stupidity: When the Romulan commander discovers the cloaking device has been stolen, it's clear someone from the Enterprise took it. This is one of the most important pieces of technology the Romulans have ever developed. Rather than take immediate action to retrieve it, she inexplicably allows Spock the right of statement, giving Scotty the time he needs to hook up the device and get away.
Stupidity: There are multiple episodes in which the Enterprise's mission of seeking out new life encounter civilizations with technology far above the Federation's, but it is never incorporated into their own.
