Over here, trainers are called either tennis shoes or sneakers. Cross-trainers are a specific type of athletic shoe, that's really the only time I hear the phrase here. First time I tried to get into a club in the UK, the doorman told me "no trainers", and I was a little confused.
But why are they called "sneakers"

About the bra incident: Small bras for girls (age 12-13, when they first start "blooming") are called training bras.
Thanks for clearing things up.