On my account there are clearly two clear boxes, yet when I try to fill it up or buy anything I cannot and it reads on the notifications : "The Bag is full"
The inventory is about to change pretty radically (we're not adding more slots, but will allow bags which have slots of their own and special containers for certain kinds of items like spices) all with drag and drop goodness.
However, the problem you were having today is (most likely) the result of a bug in a new rendering method for item graphics: the slots were probably full, but the graphic just wasn't showing up (and if the graphic wasn't showing up, neither would menus or the info boxes when you hover over the items). This will, of course, be fixed.