American law regards employee mistakes as “the cost of doing business”; an employer may demote or fire an employee for making mistakes (example: shipped the wrong product to a customer), but the employee does not owe the employer for the expense caused by the mistake.
There is no official doctrine in halachah whereby an employee is exempt from compensation for mistakes he makes.
On the contrary, the Gemara writes that an envoy that doesn’t follow instruction is not considered as though he worked for his “employer,” since he can say: “I sent you for my benefit, and not to mess things up!”
In fact, the Gemara in Bava Metzia explicitly mentions a case in which workers broke a barrel of wine, and were liable for payment according to the strict halachah (they were poor and couldn’t pay).
Yet, in the modern working environment most “mistakes” will come under categories of gerama, so that in practice most times the worker won’t be liable.