Can Combinations of Events Cause Things?

How do we deal with events in Science that have multiple causes? For instance, let's say we have a well. Let's make it a standard, residential well that provides water to a house. I'm probably the last person you want touching your plumbing, but I'm going to try this as an illustration. Let's assume that the well turns on whenever there is a need for water in the house. This need is indicated by a drop in water level in the reservoir or tank of water that feeds the house.

Now, anything could be causing this drop in water in the tank. Someone could be taking a shower. The dishwasher could be running. A load of laundry could be in the washer. Or, any of these things or more could be running in combination. There could also just be no water consumption except for a slow leak, which causes the well to run for 5 minutes once every 3 days.

For El Niño, we do not have anything as convenient as an enclosed reservoir that tells us when the system will turn on or off. Does that mean we can throw Occam's razor out? Probably not. It's really tempting to just get a bunch of potential causes and run a big, giant regression. However, I've heard an anecdote that folks (much more mathematically inclined than me) have already tried that. Instead, we must harness the mysterious power of simplification and figure out what is the water tank equivalent for El Niño.