Could I have a crack underneath my crown?

From what you are describing you may have a cracked tooth underneath your crown.  You should not be able to get your finger caught under the margin of the crown nor should you be able to move it.  Did you have a large metal (amalgam) filling in the tooth prior to having the crown placed?  These restorations are notorious for causing cracks in teeth due to the expansion and contraction properties that occur with them over time.  I would see your dentist to evaluate your tooth.  If it is deemed there is a crack you need to have the tooth extracted and ultimately have an implant placed.  Good luck.  

Based on your description of events, I would say that it is likely that your tooth is cracked.  However you would need to have to get an x-ray for your dentist to determine if this is the case.  The danger of having a cracked tooth under a crown, is that bacteria may be able to find a way into your tooth through this crack.  If bacteria enters your tooth and begins to grow, you could end up loosing your tooth.  The best option for you is to see your dentist again very soon.   Good luck and I hope this helps. Follow me if you have more questions.Sarah Thompson, DMD

There had to have been a reason to crown the tooth in the first place. If it was due to a crack or fracture, it's possible the crack remains. Sometimes the cracks extend beyond what we can see and sometimes the only solution is removal.

I think you have very good reason to be concerned about the status of your crowned tooth.  Yes teeth can fracture under a crown.  It is hard to say exactly what is going on but a radiograph of the tooth is warranted and further exam.  I would contact your dentist for a closer look.  Good luck!