Yes, but out of what principle? Out of a principle that anything that happens at work is your bosses responsibility?
I can see both sides, including yours. It depends on the bracket.. if it was broken or damaged then you may be justified in asking for an allowance. However, if the desk just has a steel bracket and that's the way the desk is designed then I don't think you can ask for money.
If you stumble and fall down the stairs at work that's your own fault.. unless the stairs are dodgy in some way.. my point is that you are expected to use a certain amount of common sense to avoid damaging yourself.