I put 60W CFLs in two lamps that had three way switches for 60W, 100W, 150W incandescents. The CFL bulbs only work on the middle setting of the switch. Why is that? Never had a problem, but my dad always said that my mind wonders.
The "old-fashioned" three-way bulb has two filiments in it, one is 60W and the other is 100W. If you use only the 60W, then you'll get 60W. Likewise, if you use only the 100W, you'll get 100W. And if you use them both, you get 160W. If you look at the base of that bulb, there are three contacts, the threaded outside, the metal tip, and then a ring that goes around between the two. The threaded outside is connected to one side of both filiments. The center contact is the other end of the 100W filiment and the ring is the other end of the 60W filiment.
A fixture set up for three-way bulbs has three contacts, the threaded outer part, one right in the middle, and another inbetween. When you turn it to the first click, it powers only the ring contact. The second click powers center contact only, and the third click powers both.
The base of a CFL has only two contacts, the threaded outter contact and the center contact.
So, when you put a CFL into a three-way base, you will not get any light on the first click because the base of the CFL has no ring contact, but should get light on the second and third clicks since both of these power the center contact. You'll get the same amount on each. However, you may not get light on the third because when you go from the second to the third click on the switch, power to the CFL will be interrupted momentarily and that glitch will cause the arc inside the CFL to go out but will not be long enough to trigger the CFL's restarter circuit to restrike the arc.
To start a florescent light, you have to apply a brief, high-voltage pulse to it. Once the arc is established, the voltage necessary to sustain it drops dramatically. You may remember when florescent lights first came out they had to be started manually. You pressed a button in and held it for a second or so before releasing it. Some had a special switch that you moved all the way to one side and held for a second or so before releasing to return to its center position to run. In either case, you were using a capacitor (called the "starter") and an inductor (called the ballast) to create a brief, high-voltage pulse that struck the arc inside the florescent tube. Then, science invented the automatic starter.
One problem was that the starters went bad from time-to-time and had to be replaced occationally. Another other problem was that the way the circuit worked, current flowed through the ballast all the time. It had to because you can't interrupt power to the arc for even a split second or it goes out. So, the ballasts got hot. Anytime you're generating heat from electricity and you're not a space heater, that's inefficiency. Another problem was that the heat also aged the ballasts which failed from time-to-time. And a problem for many years is that the ballasts were filled with oil to help insulate and cool them. That oil often contained PCBs and, once science figured out that PCBs aren't nice stuff to have around, the ballasts became hasmat stuff. Keep in mind that the heat or manufacturing defects sometimes caused the ballasts to leak. A lot of florescent lights are installed in overhead ceiling fixtures, so this resulted in hot, PCB-laden oil dripping down on people's heads.
The solution was the electronic starter/ballast found in just about all florescent lights these days. This is what's inside of that box that's at the base of a CFL. It's job is to make a high-voltage spike when you first turn the light on to strike the arc and then to keep the arc going after that with a lower voltage. The glitch, the fraction-of-a-second interruption of power, that occures when a three-way fixture goes from powering only the center contact to powering both the center and the ring will interrupt power to the arc resulting in it going out, but it may not reliably trigger the starter circuit to restrike the arc.