Factual error: The movie takes place around the time of WWI, definitely before 1923. In the carnival scene, the calliope is heard playing "Ain't She Sweet". This was the number one song ... in 1927.
Continuity mistake: When James Dean's mom writes out a check for $5000 and gives it to him, she licks the envelope first, but when she actually hands over the envelope to him, it's clearly not sealed.
Suggested correction: I noticed this, too, but as it's all one shot, it's not a continuity error; it's simply that she didn't properly seal the envelope. Not a film mistake.
Factual error: The opening scenes in 1918 Monterey show a US flag with 50 stars, and clothes inappropriate for the period.
Suggested correction: The flag shown has 48-stars. Even though I verified it by watching the scene, I know it would be impossible to have 50-stars since there were only 48 US States in 1955. And nothing about the clothing seems "inappropriate."
Continuity mistake: When Cal jumps on the train he wraps his sweater around himself to keep warm, covering his mouth, but in the nest shot with his voice over, the sweater is just wrapped around him up until his neck, with his mouth free.