The phenomenon that you describe with the copperhead bite is called anaphylaxis. The unfortunate person's immune system was exposed to proteins in the venom and became "primed" after the first bite. This involves large numbers of inflammatory cells (mostly mast cells) having antibody "receptors" (IgE) specific to that protein in the venom that wait for a subsequent exposure. The second bite results in the protein binding to the preformed antibody receptors which results in the mast cells dumping massive amounts of inflammatory chemicals such as histamine. Histamine shuts down the lungs, cardiovascular system, etc. Basically, victims of anaphylaxis die from their own body's exaggerated immune response rather than any direct effect from the poison. For example, certain proteins in peanuts can cause anaphylaxis in succeptable people.
While people's immune system can be "desensitized" thru repeated micro exposures, this is a risky process for people succeptable to anaphylaxis. People (who are not succeptable to anaphylaxis) can also develope protective antibodies to toxic components of venom. This antibodies bind to toxic molecules in venom and neutralizes them much the same way that anti-venom works. However, this also takes repeated, small exposures before any measurable protective effect is appreciated. Thus, you are likely to suffer a similar mild effect from another BR bite provided that you are not one of those rare cases who is prone to anaphylaxis.
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