Well, at least you know that the source of the 'problem' is rainwater. Now, I know you said you did the 'usual things' by keeping the gutters and downspouts clear but what you ALSO should do (since it's 20 years later) is put on some boots and grab an umbrella and go outside DURING the next rainstorm. Of course, watch out for lightening as well as watch for where the water goes when it comes off the roof and down the spouts. You might find that some water is running behind a gutter (and down an outside wall) or off the 'side of the roof' (where there isn't a gutter) or out a downspout and running right back towards the foundation. Then again, like once at my home, the water was coming from my neighbors lot!
So, since you know you have the problem when it rains, go out in the next 'heavy' rain. Also, you didn't mention about the sloping of your landscaping. The minimum sloping should be 1/4" per foot for a distance of 3' AWAY from the foundation. Of course, the more sloping and distance, the better. I know it sounds elementary to go out in a rain storm but, really, it is elementary. The VAST majority of water in the basement problems are solved by fixing gutters and/or downspouts and/or sloping and/or 'diverting' water FURTHER away from the foundation. Oh, make sure your downspout(s) aren't 'dumping' water at the foundation. You should have 3' extensions on them (at least) OR those 'decorative' pans that guide water away from the foundation.