Let's be clear: using a teleconverter is a compromise. Adding glass to an already complex and tightly controlled optical chain always means worse results than with the lens alone. Always. The only question is: how much worse? That can range from a modest and mostly invisible detraction (e.g., using the TC-14E on a 300mm f/2.8 or 300mm f/4) to a highly visible one (e.g. using the Kenko 3x on a 500mm f/4). But the results will always be at least a bit worse than, not equal to what you'd get with the lens alone. Thus, you need to pay particular attention to just what you're losing and make sure that you're okay with that.
These days of high megapixel counts and cameras with built-in “crop” abilities (e.g. DX crop on an FX body), you’re almost always better off just using the crop instead of adding a TC. Cropping doesn’t do much to impact image quality (indeed, tends to only use the best area of the lens), while using a TC always degrades image quality. On a D810, for example, yes I can get 36mp of degraded acuity with a 1.4x TC, or I can get 15mp (DX crop) without any degradation. It’s rare that I’d pick the TC over not using the TC.
Most people seek out teleconverters because they see them as the poor man's solution to a problem (more reach). But it's a fool's path to believe that's the real solution. Focus will slow, contrast will go down, acuity will suffer, and in some cases if you're not paying attention to how you handle the combination you'll get intermittent contacts between lens and body. In other words, you may end up adding problems to solve one, that of getting closer. Most people start out thinking they can rely upon converters to give them reach. They eventually realize that isn't the best solution, and they then do what they put off doing in the first place: getting the right lens, or when they can, moving their shooting position relative to the subject.
I should also point out that converters add another set of manufacturing tolerances to the equation. You absolutely need to AF Fine Tune any lens+TC combination you use. I’ve not yet found any combination where adding a TC retained the AF Fine Tune number for using the lens alone.
To me, teleconverters are only the solution when there is no other solution. If I've got my 500mm lens on my highest resolution body with me and I simply can't get any closer to the eagle's nest, then a teleconverter is the only solution. But before I got to that point I: (1) brought the longest lens I could; (2) brought my highest resolution body (so I could crop if I had to); and (3) moved my position as close as possible.
The usual push back I get from that comment is this: but that D5 and 500mm and the tripod you use with them are really heavy. Yes, they are. That's the price I pay for getting the best possible results. The minute I start thinking that a 300mm with a TC handheld is going to be the right choice, I'm introducing a huge number of compromises into my shooting, all of which are going to show up in the pixels. Indeed, I’d generally pick a D500 or D7200 over using a TC on my D810 in most cases.
Are there times when I can't carry the 500mm and big pod? Sure. But again, I don't move to teleconverters as a solution until there is no other solution. Consider my Olympus OM-D E-M1 with the Panasonic 100-300mm lens, for example. Lighter, smaller, will work on my smaller tripods, and gives me 600mm reach with better results than using my D5 in crop mode with a 70-200mm and a TC-20E III. Which would I chose? Same thing with the Nikon V3 and Nikon’s 70-300mm CX lens (see Update, below).
Like any tool in your shed, a teleconverter should be used when it's the right tool for the job. Not because you're lazy today and want to use a lighter, more convenient tool.
Update: We now have another solution from Nikon that's arguably better for some situations: a 2.7x "teleconverter" that imposes no optical penalties. It's called a Nikon 1 camera (e.g. V1, V2, or V3). When you use the FT1 adapter (which has no glass in it) on a Nikon 1 camera, your effective focal length is 2.7x greater than the marked focal length of the lens. The FX 70-300mm VR lens becomes a very interesting 189-810mm effective lens on a Nikon 1 with the FT1. Better still, Nikon now has a dedicated 70-300mm CX lens for the Nikon 1 system, and it’s an excellent lens, even better than the FX 70-300mm with the FT1 adapter. The drawback to this approach is that the Nikon 1 cameras use a small sensor (1"), and thus you're restricted in dynamic range. That's generally fine for daytime shooting. At 810mm effective you're at f/5.6. The Sunny 16 rule tells us that we should be at ISO 100 at 1/800 in bright daylight. The problem is that if we have to bump up the ISO on a Nikon 1 because we're not in bright sunlight and/or need a faster shutter speed, we're going to start seeing more noise due to the small sensor size. Still, from a practical standpoint, most people looking for that "inexpensive wildlife lens" would be better off avoiding a teleconverter on the lens they've got and just picking up a V2 or V3 and either the FT1 or the dedicated 70-300mm CX lens. When Nikon had their fire sale on V2's, that was almost cheaper than buying the TC-20E!
Update 2: With Nikon’s introduction of the 200-500mm f/5.6E lens, we got another solution that’s arguably better than using a teleconverter: put that new lens on a D7200 or D500 body. Because of the DX crop, you’re now getting 20mp or 24mp at an equivalent of 300-750mm. If you need more pixels and more reach than that, you’ve got problems, my friend, lots of them. This solution holds up better in lower light than the Nikon 1 + 70-300mm CX lens, and is thus my preferred cut-down approach. On the other hand, the 200-500mm f/5.6 is not a small, light lens. It’s right at the edge of what is hand holdable.