When Kirk & co. land in the 20th Century, he orders everyone to remove any Starfleet insignia and the like. But when Chekov is captured on the aircraft carrier Enterprise, a security officer reads aloud from an ID that Chekov had on him, an ID that reveals Chekov's real name and identifies his affiliation with Starfleet and the United Federation of Planets! Just the sort of thing you want to carry on you during an undercover mission, the 23rd Century equivalent of your driver's license.
You'd think that something like this would've been covered under Kirk's order, and that Chekov would've have saved everyone a lot of trouble if he didn't have it on him.
So what do you think? Was Chekov disobeying an order? Should Chekov have left his ID on the Bounty or not?

So what do you think? Was Chekov disobeying an order? Should Chekov have left his ID on the Bounty or not?