I dont have particularly good wrists. But I am determined. Is it possible to get a good chest by working hard, or is it predetermined that I cannot have a good chest because of my wrist size?
TLDR: Yes, you can certainly have a strong/big chest, or any other muscle you are looking for with hard work and the right diet even with small wrists. You might not have the same potential as the top couple percent of people, but that is just a reality of life for most people.
In this study they examined the differences of Total Body Fat (TBF), Fat-Free Mass (FFM), Bone Mineral Content, and Bone mineral Density in relation to peoples joint breadths. You might have heard someone's interpretation of this quote leading you to ask this question.
Wrist breadth is potentially the best discriminator of an association between frame size and amounts of fat and muscle, independent of stature. Broad wrists are negatively associated with TBF and positively associated with FFM and vice versa.
A common interpretation of this is that the smaller your wrists, the lower your genetic potential likely is for large muscles. This means it might be harder for you to put on significant amounts of muscle overall, but it does NOT mean it is impossible to get seriously strong or muscular just because your wrists are small.
Anecdotally, I have tiny wrists, and my chest is definitely my strong point.