Factual error: When Leavey begins her training at Parris Island in 2001, she and all the other Marines are wearing MARPAT (Marine Pattern). In reality, MARPAT would have just been designed, and not manufactured until early 2002. The standard camouflage for Marines at the time was M81 Woodland.
Factual error: Performing for Queen Victoria is the famous opera composer Giacomo Puccini. This happens during her sojourn in Florence in 1888. Puccini was born in 1858, so he was barely thirty years of age, but the actor portraying him, Simon Callow, is in his late 60s (and it shows). Moreover, he is presenting his latest creation "Manon Lescaut"; in 1888 he hadn't even started working on it.
Factual error: Since the movie was set in the 60s, David Senak couldn't have gotten in trouble for shooting the man running away. In the 60s, the police could legally shoot any criminal running away. This didn't change until Tennessee v. Garner in 1985.
Factual error: Tony writes a "postcard" on the back of a picture to Adrian and Veronica to an address in Cambridge including the postcode. This scene occurs in 1967 (we see the date on the letter he is responding to) but postcodes weren't in use in Cambridge until at least 1970.