We are all no doubt familiar with the phrase "with bated breath," but is it ever used in other contexts?
Yes. Aside from its meaning within the idiom "with bated breath," bate as a verb can mean 1) to moderate or restrain, or 2) to lessen or diminish, among other definitions.
To give an example, as you can see in this dictionary entry, one can have bated sentiments such as bated enthusiasm or bated hopes:
An example sentence here would be "After waiting two and a half hours in line for the Empire State Building observatory, we finally trudged into the elevator with bated enthusiasm."
It would seem that many things can be "bated," so long as it makes sense for it to be moderated, restrained, lessened, or diminished. There is even a proverb that goes "In the report of riches and goodness always bate one half."
In historic England (a long time ago) there was a form of entertainment called bull or bear bating. This was were a pack of dogs were set to attack a bull or bear, it was said then that a certain bull etc, was 'bated'.
From this came the English expression to 'bate' someone, that is for a group of people to deliberately annoy someone, as the dogs did in bull or bear bating.