Could your gum come apart from your mouth?

Periodontal problems: If your question is about gingival tissue around your teeth looking as if it is pulling away, then there is a very strong chance you are suffering from periodontal disease. This requires immediate attention by a dentist, and if left untreated can result in the loss of your teeth.

Gingival recession?: I think that you may refer to your gum receding from the neck of the tooth. This situation is caused by gum disease which bone is loss and tissue follow the bone and recede, heavy brushing with hard or medium hard bristle toothbrush (always use soft) causing gum injury and recession, and finally localized food impaction causing bone loss and gum tissue recession.