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Even when I dump a pale of water into the bowl, the toilet still will not flush unless I plunge it.

I do not think there is anything wrong with the toilet and am wondering if the plumbing may be the problem.

I have look into the common problems but cannot explain why it sometimes drains fine and other times overflows.
Is it possible that too narrow of drainage pipes could be causing this? Could it be that there is a vacuum buildup preventing the flushing?
The drainage lines were recently installed and so are clean. My concern may be that part of the drainage line may be too narrow at 2 inches.

I cannot explain why it sometimes flushes just fine.

Our home sits on a downsloped front yard, and heavy rain often pools in the front of our house and apparently pushes in through our stone foundation and into the finished basement. We recently had a trench and sump pump installed, which normally takes care of the leaks on two walls during average to moderately heavy rains. But during the heaviest of downpours, there's just so much water that it flows into new cracks in the foundation and is possibly forming new openings -- openings that aren't surrounded by that new and expensive trench. I'm considering installing a French drain in my front yard, but I'd like to be sure this will solve the problem, or at least remedy it. It's getting tiresome to have to vacuum the water from the carpet in my basement each time it rains heavily. I hoped to have this past us when we had the trench and sump installed, but it's still flowing in. For those of you who've had similar problems or have experience with drainage issues, what should I do next?