Flattening would reduce the file size, but I think we have to clarify what it is we mean by a "one layer" document. Because there are two kinds of "one layer" document in Photoshop.
If your Layers panel only says "Background" then you have a flattened document. There are no other layers, and because it has a Background, there is no transparency, so no overhead data is set aside to handle transparency. You know this document is flattened because the Flatten Image command is not available.
If your Layers panel says anything else — including one layer that says "Layer 0" or any name other than "Background" — then it is not a flattened file. It has one layer with the potential for transparency, so there is overhead data set aside for that transparency. In other words, it's a layered file with just one layer. You will see the Flatten Image command available for this document because it isn't flattened. This type of "one layer" document always has a larger file size than a truly flattened document. (This is what Chuck's example demonstrates, by the way.)
Also, in case this has something to do with it: Don't confuse the Layer > Flatten Image and Layer > Merge Visible commands. The result of Merge Visible is a one layer non-flattened document, so it will still have a larger file size than if it was flattened. To truly flatten a one layer image, either use Flatten Image, or select that last remaining layer and choose Layer > New > Background From Layer. Either way, as soon as your entire Layers panel is reduced to a single item called Background, you won't get a layer-related file size warning and your file size will finally go down.