Can an x-ray miss a cavity under a crown?

Yes, a cavity could be under a crown & not visible on xrays. Xrays are 2 D & could be taken from different angles. For that reason, although xrays are very helpful but they are limited.
The other issue is that the opacity of the crown is most probably blocking whatever is going on underneath. I would recommend to see an endodontist to see if that tooth is the one causing your pain.

Hi, thanks for sending your question. Often when an xray is taken from and existing crown the inside portion of the tooth within the crown is not visible, therefore if there exists a cavity, will not be visible. Only caries at the margins or below will be visible on the xray. The dark area along the gum line may be the metal portion of the crown if your crown has metal. It would be best to examine the tooth carefully and check the areas. All the best.

Hello!
There are 3 things i would like for you to take into consideration, first, is your crown metal free? most porcelain crowns have metal on the inside, and this makes a dark line show up along the gum line after some time of placing the crown. second, do you have a root canal in this tooth? if not, it is quite possible for the tooth to be experiencing some discomfort after some time of a crown placement even without a cavity, and third, yes, the material that is used to manufacture the crowns will most likely wont allow for an xray to show a cavity if there is one, but this is as you can see not the only reason for pain or discomfort. 
Hope this helps, good luck

Hello,
I would have to check the x ray to see what actually is that dark stain along the gum line and the actual cause of the pain.
Dr. Franco
DentArtist Clinic