Is it possible to fix a hole in a tooth near the gum line when the rest of the tooth is intact?

Based on your description of events, I would strongly encourage you to get a dental examination as soon as possible.  From the sound of things, it appears that your tooth is decaying and you will likely need the cavity fixed and a crown placed to protect the tooth.  Your alternative is to do nothing and wait for painful symptoms, which will probably lead to a root canal or extraction.  Good luck and I hope this helps. Follow me if you have more questions.Sarah Thompson, DMD

It may not be your tooth structure, it may be a chunk of bacteria build up called "calculus".  You don't know many times I've gotten patients in my office claiming they broke their tooth, but in reality it was a chunk of calculus because they haven't been to the dentist in years.  Go to see a dentist ASAP to have your oral health evaluated and get that area checked.  If it is indeed the tooth structure itself that broke off, then absolutely we can fix that in dentistry.

It can be multiple reasons why this happing. From routine cavity to abfractions and abrasions. It is easy to fix the problem using bonding. The question is why and what is happening. Please consider a dental consultation, so that he can examine your mouth and diagnose the problem properly before fixing it.

Yes, a hole or cavity can happen next to the gum line and can even extend under the gum line. Those cavities can be treated with fillings. Get a dental exam to determine the condition of the tooth and to start treatment. Good luck. 

It is probably either a gum line cavity (decay) or something called an "abfraction" cavity (caused by clenching or grinding your teeth).  Either of these two types of cavities can usually be easily be filled with a bonded tooth colored filling material.  If keeping your tooth and not getting a toothache is important to you, then it is best for you to see your dentist to have him/her actually diagnose exactly what is going on with your tooth and recommend appropriate treatment.