Why does my dimmer only dim a little, then cut off?

So I bought dimmable light bulbs (CFLs, 60W equivalent, max rated for the fixture) and a dimmer switch for the overhead light in our bedroom. The fixture is a 2 bulb unit. Once installed, everything works, but you can only go about halfway down on the dim before it cuts out...and it's not very dim. Just before it cuts out, it goes down to one bulb, which will cut off on its own within a few minutes. Then you have to go back to full power for them to cut back on.

I'm confused as to how to fix this problem. Is it because it's a dual-bulb fixture? Would I get full range if I went down on the wattage of the bulbs? Is this all a product of the fact that I'm plugging the fixture into the wall, which is attached to the switch? My limited knowledge of these things tells me that the latter shouldn't matter, but I really thought this setup would just work as is, too.