Most of the time, when a person thinks their dental implant fell out, it's not their actual implant.

It's typically their healing cap. 

There are many sizes and styles of healing caps. 

A healing cap, also known as a healing abutment, helps promote soft and hard tissue healing around an implant. The healing cap also protects the main part of the implant from plaque and debris accumulation. The cap is screwed into the top of the implant. They are placed at the time of implant surgery and usually remain in place until the crown and abutment get placed. They are identified by having a smaller and thinner screw part on the bottom. Healing caps are very small and are often smaller than a pinky fingernail. 

Below are images of healing caps, and then the  images after that are images of dental implants.