Even if you no longer work for the same company you are eligible to receive workers' compensation benefits. There are several issues with the facts as you have stated them. In Texas, you must report a carpal tunnel injury to your employer within 30 days of the day you knew or should have known that you had a work-related injury. You must also file a claim with the Division of Workers' Compensation within one year from that date. There are some exceptions, but that is the general rule. If you did report the injury to the employer back in 2008 and just never pursued it, that would make it easier. To see if you meet any of the potential exceptions, you should contact an attorney to discuss the details of your situation.
The short answer is that you may file a claim, but certian obstacles may allow the employer to contest or dispute your claim. Namely, when did you first seek treatment for your ailment and when did you first realize that the problem may be related to your work. Also, whether you have worked since being terminated from that position and in what capacity is an important question to answer.
I would highly suggest you consult with an experienced workers compensation attorney in your area to discuss these issues.
Carpal tunnel syndrome not due to a trauma or single injury is often considered an occupational disease or repetitive stress injury. The workers' compensation law governing benefits for this condition varies by state. However, a workers' compensation claim may generally be made for this. Whether your particular claim is still viable depends on a number of factors including when you first knew or reasonably should have known that your condition was due to your employment activities (for example, when did you first seek treatment and did your doctor tell you your problem was work related?), did you report the carpal tunnel syndrome due to your employment as work related and did you file a workers' compensation claim within the statutory time limit. The responsible employer and carrier would have to be determined depending on your work history. An experienced workers' compensation attorney licensed in your state will be able to help you sort this out and determine whether you still have a viable claim.
This answer does not constitute legal advice nor does it create any attorney client relationship. You should consult with an attorney licensed in your state who is experienced in the area of law involved in your question.