Of course a person doesn't die immediately from an overdose, there are previous effects to show that you're overdosing. Like from Med-X withdrawal, your stats lower and things get foggy from time to time. The same concept can be used, and just the same way if you go under water - if you stay too long you begin to rapidly lose health. It can be the same effect, giving time to get to a doctor if you can. Some things to fix up an overdose could be Fixer. Although it only removes addiction, what's to say it can't stop an overdose? Kind of like a shot of adrenalin can curb an allergic reaction. It really adds good immersion to the game, at least I think so.
What do you guys think?
EDIT: I've been addicted to Med-X thanks to my Stealth Suit for a while and haven't really felt the effects of it. It doesn't really 'hit me hard' like it does in the real world. An overdose reaction to taking a lot of Chems in the game would really shake things up a bit. Plus, it's not like we haven't heard of overdoses in Fallout before. Boxcars in FNV can overdose if you give him enough Med-X.