Where is the water coming from?

Here's my mystery: I have a house built in 1937. The downspouts used to connect to crockery-type pipes (sealed with concrete around the spout) at each corner of the house. I also have a basement floor drain that does not appear to have a trap -- in fact, it appears to just be over a pipe or a small open area. I lived in the house for 12 years with nothing other than an occasional water seep around the foundation, no big deal.
A couple of years ago, the basement flooded with several inches of water for the first time. I disconnected all the downspouts and ran them away from the house. ALL downspouts are now disconnected. A few months later, the basement flooded again. A plumber suggested putting in a sump pump (this was after a failed attempt to snake the line -- he ran into a blockage he couldn't clear and speculated the pipe may have collapsed or been blocked by tree roots too big to get out), which I did. I've had no problems when the sump is running, only when I've lost power have I had a water back up again (during rain only).

But here's what I don't understand. Last night, it was lightly raining. I went to the basement and could hear water. Listening at the sump pit, it was coming from there. There was no water coming in from the weep holes making the noise. There is an old drain pipe that used to be part of the outside drainage system that bisects the pit. The plumber ran the pipe into the pit "just in case" -- even though all the downspouts are disconnected from the system (and sealed off). There is no water running out of the pipe, but it was where the sound was coming from.
So it seems that there is still a decent amount of water (about the amount that would come out of a downspout during a light rain based on the sound) flowing through the system -- but where the heck is it coming from?
Is it possible that my neighbor's old house is somehow connected to my system as well? Is it just ground water seeping into the old pipes and running through it (but where is it going? It's not going into my pit).

thanks
Todd

While illegal today, those old pipes might have run to the city storm drain and you're hearing water flow through there?

I'm not sure I quite understand what you mean but...

...even if the downspouts were disconnected those pipes would probably still pick up a fair amount of groundwater. Drainage pipes are often not "sealed" like water pipes, so water can seep in around the joints or though other gaps underground in the system.