most people getting asked that question aren't in near as much pain as they could be, and they don't have much to gain by inflating their answer.

Maybe they think that by inflating their answer they gain, on the margin, better / more intensive / more prompt medical service. Especially in an ER setting where they may intuit themselves to be competing against other patients being triaged and asked the same question, they might perceive themselves (consciously or not) to be in an arms race where the person who claims to be experiencing the most pain gets treated first.