Tiles or carpet with rubber backing or vinyl flooring are about the worst things to put on a basement floor. There is a constant vapour drive upward. Normally, that vapour is dispersed in the air and noticed as high humidity. But when you put those impermeable materials on the floor, the vapour is trapped and may condense into liquid water. And then run and collect in the lowest spots. A dehumidifier is necessary in most climates to keep the relative humidity (rh) below 50%. Be sure to keep yours running. If, with the tiles gone and the rh controlled, you're still getting water accumulation, it may be from the leaky walls. And even if the water comes up through the floor, it may simply be from water accumulating on the walls with no place to go or overwhelming the ability of the weepers to drain it. For instance, during heavy storms a homebuyer of mine complained of water running under the basement floor. You could hear it and actually see it around the basement drains. Turns out the source was the downspouts. The water ran down the foundation wall and accumulated in the five-inch gravel bed under the concrete floors. Since then, I always run the downspouts into underground drain pipes running far out into the yard. The plastic on concrete test has long been outmoded by moisture meters. You're almost always going to get moisture under the plastic because of moisture drive. This post was edited by worthy on Thu, Sep 26, 13 at 18:32

Interior systems may keep the living area dry. But they do nothing to stop the deterioration of the wall and the underlying soil from the flow of water. I would only use an interior system where: 1) there is no access to the exterior, as in homes built on or near the lot line next to similarly situated homes; 2) the water is coming up through the basement floor; 3) foundation walls cannot be excavated without risk of collapse, e.g., rubble walls; 4) it is in conjunction with exterior water control measures.

Where I live a French drain has no pipe at all. It's simply a drainage ditch that is filled with gravel or drain rock, and then sometimes covered over with soil. If that is the type of French drain you have, it's entirely possible that the drainage stone has silted up over the years.

There are several good articles on the net about how to finish the basement walls, and why it is not the same as the rest of your house. You need to stop water penetration into the walls, and you already plan to add waterproofing on the cement. On top of that you should install closed cell foam board that will seal out water and insulate the wall from top to bottom. Also insulate the sill plates. The foam board should be 1' minimum, and 2" is better. Then frame a 2x4 wall inside the foam board, using pressure treated wood sill plates ,and finish with sheetrock. This site has a good explanation. http://www.greenbuildingadvisor.com/blogs/dept/musings/how-insulate-basement-wall Do not install any type of vapor barrier on the habitable side of a basement wall, since that would trap moisture in the wall. This is the opposite of an above ground wall, where the moisture barrier usually goes on the habitable side. Make sure your contractor understands the difference. Bruce