I've grown my own yeast out of bottles before. Basically, all you're doing is making a very small yeast starter and pitching the dregs of a bottle into it and growing it up from there. I usually start with 25ml - 50ml out of a bottle and then step it up a couple of times until I'm making a 1000ml starter for a batch of brew.
I've had luck doing this with English-style ales and Belgian-style ales, both.
You'll want to make sure that you can sterilize the mouth of the bottle before you pour out of it, since that's the place that's most likely had some sort of other bacteria get to it. You can swab the mouth of the bottle with a small amount of isopropyl alcohol and burn it off for quick sterilization.
A few important things:
1) You must do this from bottle-conditioned beer. If the yeast has been filtered out prior to packaging, there will not be enough yeast in the solution for you to grow.
2) Not all bottle-conditioned beer is bottled with the same yeast that it was originally fermented in. Many breweries use lager yeast for bottle-conditioning, since they can then put the bottles into cold storage or cold shipping and it will still condition, ferment, and carbonate, just more slowly.
3) It has to be fairly fresh beer, as the yeast will suffer quite a bit of stress under packaging conditions (high pressure, not optimal temperature). You may not be able to culture yeast out of something that's been fairly aged, or what you do culture out may be a mutated strain.
4) Over time, yeast strains will mutate. Most commercial breweries will re-pitch yeast batch to batch for a few generations (6 - 10 is normal, I think), but at some point, if you're looking for consistency from your yeast strain, you'll want to wash your yeast and grow a pure culture up from scratch again, and this will be a pain in the ass without some sort of lab. Mind you, in a homebrewing environment this may not be much of an issue, so long as the mutations you get are pleasant.
I'm on my 4th generation of yeasts that I've grown out of bottles, now, and everything still tastes great. I have a yeast library of four yeasts that I've grown up from scratch - two belgian yeasts, one english-style (which I think is Whitbread), and one English-style combination (two bottles of different ales together).
That said, I still buy yeast for probably half of my batches, just for variety in flavor profile.