A fish should never suddenly release enough ammonia in a cycled tank to bring the nitrites to 5.0 ppm. You might see a slight reading, but not significant, even if a few fish are added or more food is being given. Once you get a certain amount of bacteria colonizing your filters they will multiply and remove nitrites and ammonia fairly quickly. This is why you can "quick cycle" aquariums by adding established media, even in small amounts, to new uncultured media, though oxygenation, temperature and pH, among a few other factors, will effect how quickly your new media is colonized.
The only way you would see such a rise in nitrites is if the tank was never cycled or there is not enough media to house sufficient bacterial colonies, or if something with the water chemistry is impeding the bacterial growth/function. (I don't see the latter as the case if the pH has always been stable).
Most likely this tank does not have enough media to grow sufficient bacterial colonies. The fact that the ammonia has always been above .25 indicates this (given sufficient media it should have always been 0), and if you have always had nitrites either you're severely lacking enough biomedia, somehow killing it repeatedly, or the tank was never fully cycled.
What kind filtration is on the tank and how much/what type of media do you have? Do you ever remove the media and replace or "clean" it, and how often? I can pretty much guarantee that the filtration on this tank needs to be upgraded and/or more biological media added. Once you add more media the issue should be resolved fairly quickly as the bacteria on the cultured media colonizes the new media.
For now you need to do several water changes to remove the nitrites and get them down around .25 or less.