Did any events in the movie Saving Private Ryan actually take place?

Several of them did: most obviously the landing at the beginning of the movie, known as D-Day. The story of Private Ryan's 3 brothers being killed in combat was loosely based off the story of the Niland Brothers. To quote: "Of the four, two survived the war, but for a time it was believed that only one, Frederick Niland, had survived. Frederick was sent back to the United States to complete his service and only later learned that his brother Edward, missing and presumed dead, was actually captive in a Japanese POW camp in Burma. Steven Spielberg's film Saving Private Ryan is loosely based on the brothers' story."