Probably. Your employer can also seek what is called liquidated damages if those are set forth in the non-compete agreement. Liquidated damages are a set sum that the employer and employee agree to as damages if the employee breaches the covenant not to compete. Not all liquidated damages are enforceable under the law, however. This too depends on the facts of each case and the law of each state.
Additionally, the employer can seek any actual damages or losses which they claim have occurred because the employee left in violation of the covenant not to compete - this could include lost profits from customers, the loss of secret employer information and similar losses.