It seems like anytime I kill a barbarian encampment in Civilization 5 another encampment pops up somewhere else in my territory. Is there a certain number always on the map or am I seeing patterns where there are none?
Don't belittle making patterns where there are none. It's a favorite unintentional past-time of the human race! (See: Gestalt theory).
However, Barbarians can only spawn in the "fog of war", as it were. It's completely possible that if certain areas are the only unclaimed areas on the map (meaning that all barbarian spawns must happen from within the area) and you have only a small military force ping-pong-ing back and forth between barbarian incursions, it's fully possible to spend many a fruitless turn chasing down recalcitrant barbarians.
Barbarian spawns can be eliminated completely when all of the passable land tiles are within view-distance of a civ's cultural borders. Without any area in the fog of war, the barbarians effectively die out.
Actually you are not imagining this. There is a logic in the game based on number of existing camps vs non-water fog tiles that determines the chance for a new camp to spawn. Destroying a camp is likely to significantly increase chances for another spawning on the next turn.
The exact mechanics has been thoroughly investigated in the past. This forum thread has further details: https://forums.civfanatics.com/threads/barbarian-spawning-logic.544660