I'd say less than 1% of the time, if ever.
When you're looking through the backlinks of your competitors, you are never going to be able to get all of those links that they have. In fact, the acquisition rate is often pretty low (but still worthwhile, certainly). Some of the sites will be shut down, some will have linked specifically to them for a good reason and aren't going to link to you just because you ask. So at best you'll have some percentage of what you see they have.
But of course a bigger issue is that you are likely only seeing a small portion of their backlinks, so they're also benefiting from a bunch of links that you can't even see.
Mining the competition's backlinks is a good tool in the prospecting toolbox, but you're also going to have to find a lot of links of your own as well in most cases, to have a shot of ranking up near where the competition ranks.