Do I have a case if I was opening a bottle with an opener and the glass broke at the top causing a 1/2 inch cut on the inside of my thumb?

As a general matter, there is no compensation or recovery in a case like this unless one of the products (the opener or the bottle was defective); if there was no defect, then you cannot recover compensation simply because you were hurt. (Often, people are hurt without any fault or liability.)

Furtheremore, product liability cases can be expensive to bring, since you typically need tests or experts to establish the defect. However, the most you can recover is an amount equal to any unreimbursed or out of pocket medical costs and lost wages; you would not be able to recover anything for pain and suffering, even if the product(s) was defective, unless you suffered some significant injury (e.g. permanent partial loss of function of the thumb). Thus, a lawsuit would not economically be worthwhile.