This is more common than you might think, particularly on older iPhones. I co-own a repair store in Austin, TX and we get several of these every month--usually on older iPhones that have been dropped "just right". (Not saying yours was the fault of a drop, but if you leave it plugged in 99% of the time, that was probably your issue.)<p>Anyway, it's stupid easy to replace these batteries--they're just glued in, usually, so you just buy a replacement battery and back cover, use a tool to pry up the battery, place a new battery in along with some adhesive to hold it down, and pop the new back cover on. One pro tip: Buy at least one more new battery than you need, as some percentage of them, no matter where you buy them from, are defective.<p>We do these in-store for iPhone 4/5 in under 10 minutes--I don't doubt you can do something similar on the N4.