His glasses had slipped out of his pocket and fallen in/into the crate.
Is 'fall in sth' good English too?
Thank you in advance.
Not as good as "into". If he was standing in the crate when it happened, "in" would be correct.
Sometimes there is a definite difference, eg. if your grandfather fell in his room, he was in his room when he fell. If he fell into his room, he wasn't.