Generally, no, with limited exceptions. When you are hurt on the job, you usually recover compensation only through workers compensation. Employees in all states are covered under workers compensation rules at the federal or state levels. Thus, you usually cannot file a personal injury lawsuit in addition to taking workers compensation. The exceptions to this rule are:
Your employer tried to intentionally harm you. For instance, if your supervisor punched you in the face, you could theoretically sue for damages in addition to taking workers compensation. But the punch in the face must have been intentional; if it was an accident, it was not intentional harm and does not qualify.
Fraud: Someone at the job lied to you and you suffered an injury
Defamation: Someone at work said something untrue about you that caused you harm
Intentional infliction of emotional distress: You were emotionally traumatized on the job by extremely bad behavior
Your employer does not have enough workers compensation insurance. If your employer did not carry enough insurance to properly compensate you, it is possible you can file a lawsuit to make up the difference.
Another major exception is if you are injured at your place of employment by a third party. If the injury was caused by defective equipment that the other company produced, you can file a separate lawsuit against that company.
If you are eligible to file a lawsuit, it is important to consult with a personal injury attorney as soon as you can. Your attorney will be able to confirm if you can file a lawsuit in addition to seeking workers compensation. But most cases of workplace injury do not allow for these types of lawsuits.