Check the output pipes. My sump pump refused to work once. It turned on, but I checked the power draw - about 150% of what it should have been. Something seemed to have gotten stuck in the output pipe, though I never did find out what. I just loosened the connection to the pump, a bit of water spewed out from pressure that the pump had put there, and that seemed to clear out whatever the problem was. The pump then went nuts getting rid of what water it could.
The possible solution to the flooding has been this:
1) A hole was cut in the foundation's frost barrier at the back of the house, which leads into a bed of crushed stone. The contractor we hired said that the frost barrier can have the side effect of trapping water underneath the foundation. It flows down underground from the front of the house, seeps underneath the foundation, and is then unable to flow out the back quickly enough. Its solution: push its way up through cracks in the floor as well as through the sump. During heavy rains that persisted for more than a day, the sump pump would run about every 30 seconds for 15 seconds, the water just filled in so quickly.
This hole, which is about 6" below ground level - at least according to the way our particular yard was landscaped, this will vary on your building - has a pipe which directs the water away from the house, so as to drain this subterranean "pond."
2) A trench dug around the front of the house, which is where the water comes from originally. This trench is 8-10' deep at its deepest, about a foot wide, and goes as far down as does the foundation. It runs along the front yard parallel to the house, and then curves around the side. At the bottom is a perforated pipe to allow water in and give it a way to flow easily. Above that is filter fabric, then several feet of crushed stone, and finally topsoil and grass.
The trench works to try to prevent water from even making it as far as the foundation. The water can soak through the dirt until it hits the crushed stone, at which point it simply falls down. It finds the perforated pipe, and flows freely through that, eventually meeting with the pipe from the back of the foundation, and then to the edge of our property.
In previous years when it's rained very hard for a long time, in addition to coming in through the floor and the sump, water also came in through cracks in the front foundation, which made the sump pump slightly useless, as it was only attacking one of the water's many inlets.
It's been there a year at least, but I still don't know if it works, as it's not yet rained enough to let me really say for certain that the trench is doing its job, and that the basement will not ever flood again. But it all looks pretty good in theory, and the sump hasn't been so much as damp since the trench was made.