Contact (1997)

Continuity mistake: There's a scene where Ellie sleeps with Palmer, then gets up quickly to go to the lab. In her haste, she departs whipping on a shirt with no bra. If she had a bra inside that tight T-shirt the contours of the cups and straps would be visible and we would see her fitting and fastening it, instead she steps outside where suddenly it appears she is now wearing a bra.

Ellie Arroway: I work for a project called seti.
Palmer Joss: Search for Extraterrestrial Intelligence? Wow, that's out there.

Trivia: Filmmakers George Miller and Francis Ford Coppola both sued Warner Bros. over Contact. George Miller sued for breach of contract (as he was the original director before being fired and replaced by Robert Zemeckis), while Coppola sued because he claimed that he and Carl Sagan (the writer of Contact) had already developed the premise for a TV show in the 1970's which was never produced, before Sagan later used the idea for Contact in 1985. Both suits failed - Miller's firing was within contract and perfectly reasonable, and Coppola was dismissed (twice) because he had taken far too long to sue the company (if he sued when Sagan began working in the 80's, he may have won, but he waited until after the film's release in 1997 to sue).

Question: How did they film the scenes where real historical figures (President Clinton, for instance) made speeches and comments they didn't make in real life?