Although great pains were used to disguise the name of the country in the film, Dr. Ferguson mentions that Pres. Farrago should go to a neurological hospital (which he mentions is just across the border) in Chile - which narrows the country down to either Argentina, Bolivia, or Peru. Other hints include: Peso/Dollar law, the blankets worn on the train, license plates on vehicles.
Director Richard Brooks was as the Santa Anita race track where he met Cary Grant and struck up a conversation about this film; Brooks wrote the story but also wanted to direct but none of the studios would let him, but Grant asked for a copy of the script and loved it - so much so that Grant himself went to MGM and said he would love to do the movie but only if Brooks was the director.
Originally banned in Mexico, Central and South America.
The original story was about the doctor's daughter getting kidnapped and no wife, but once MGM had Cary Grant cast they insisted on a re-write so that Grant would have a love interest in the film.