I'm no architect, but is this a stupid idea?

It is a great idea and an inexpensive method to create shelving. The weight may become too much for the filing cabinets if you put too much on the shelves. The cabinet(s) might begin to bend and at some point give way to the weight. I would simply cut a piece of 3/8" or 1/2" plywood the same dimensions as the top of the filing cabinets and put them on top and then put the blocks and shelves back in place. I think you'll be good for a long time to come without having to worry about a collapse. I hope that helps.