That's the question I had in the attempts to extend an already powerful 50x optical zoom on my Canon SX50 HS. The results aren't very impressive.
I bought two teleconverters for testing. The cheap one is a Vivitar Titanium 2.0X, at about US$40. The expensive one is a Raynox DCR-1540PRO with 1.54x zoom, at about $200. Unfortunately, the latter had been discontinued a long time ago (it came out at least in 2003), so I had to buy a dusty used one at half the price.
The collection of crops above really tells it all. The cheap Vivitar is trash. Completely unusable, chromatic aberrations and all. It works okay at low zoom, but that defeats the purpose.
Meanwhile, the expensive Raynox has been advertised as "for 25x or higher zoom" and "340lp/mm at center", so I was expecting better. Well, the colours are fine but it doesn't come into focus as well as I would want it to. Even at the best spot (seems to be at lower center region, not outright center for SX50 HS) it doesn't really add more detail. This might be usable for video mode, but not much value for still photos.
As for the actual multiplication, close (~5m) shots got about 1.3x for Raynox and 1.4x for Vivitar, getting roughly about half the spec. Far (~250m) shots are better, at about 1.52x for Raynox and 1.8x for Vivitar.
If anyone's searching around to see if one needs a teleconverter for a SX50 HS, I can easily tell you "don't bother".