Technically, there is no limit to how much power they can run at. The issues with capacitive droppers are largely around safety and longevity. Safety because they provide no electrical isolation from the supply so the onus is on mechanical design to ensure a sufficient barrier exists between live parts and any environment at risk. Longevity because capacitor tend to change value over time so the operating current may not be stable in the long term. Consider that capacitors tend to drop in value and LEDs lose emission as they get older so together they both work to reduce luminoscity.
There are other considerations too, for example, the 'Z' referred to in post #5 depends on frequency so different values of capacitor would be needed in 50Hz and 60Hz regions and of course there is the problem that Z may be significantly lower if harmonics or other signals, including impulses are present on the supply.
Personally, I have used series class X2 capacitors in small LED supplies but also included over-voltage proection, fast-blow fuses and a resistive element as well. I have only done this in the most extreme low cost aplications and where the current is relatively low. If you tried it at high voltage and high current I think the cost of large capacitor will probably outweigh the cost of a small SMPS and not provide the stabilization an SMPS can provide.
Brian.