Numbness is a non-specific symptom caused by disruption of the function of a sensory nerve-- in this case, the so-called inferior alveolar nerve, which supplies the teeth, gums, and lower lip with sensation.
Disruption of sensory function can occur for a number of different reasons. The more mundane explanation might include a chronic infection
in a lower tooth-- most likely a lower premolar
(3rd or 4th tooth back from the midline). If one of your "really bad teeth" is a lower premolar on the side where the numbness occurs, this is the likely source of the problem, since the root tips of these teeth lie very closely to the nerve as it emerges from the lower jaw bone and runs into the lip.
However, if the numbness cannot be traced to a particular tooth, there are other possibilities-- most notably some persistent infection with a "neurotropic" (i.e., nerve- affecting) virus, such as herpes simplex or herpes zoster. Usually these viral infections also manifest with visible lesions, such as cold sores or other blistering lesions, although the viral infection may be active without any visible signs.
Although there are neurological symptoms such as numbness that might occur with more global nerve disorders (such as multiple sclerosis or diabetic neuropathy), these disorders are seldom isolated to a single nerve such as you describe; as such, I don't think there's cause for worry in this regard. However, if a thorough exam by your dentist fails to discover a reason for your numbness, you should consider seeking a consultation with a neurologist, if only to cover all the diagnostic bases.
Hope this helps...