Is it possible to have cold sensitivity on a crown?

Several reasons. The most common problem is the crown is not integrated into the bite and requires adjustment. The second reason is that the procedure caused trauma to the central tissues, but this should regress with time using ibuprofen. The third reason could be that the tooth, which was probably badly damaged requiring a crown could have central tissue inflammation that may not improve and could worsen.
Sometimes. It is possible for this to happen. Every dental procedure is irritating to the body. I seal the inside of the tooth immediately after I finish shaping the tooth. Then, I take impressions, usually a scan to make the crown while my patient waits. If everything fits properly, and the bite is right, this generally occurs very infrequently today.