The question you really need to ask about the search for the perfect IV is- is it worth it? That depends on exactly how far you’re going to get that IV, and how close to an absolute 31-only value you’re reaching.
Because each non-bred pokemon is randomly generated and each bred pokemon is partly randomly generated (more on that later), the basic chance of a pokemon having 31s in both defenses, speed, HP, and one of the two attack stats (assuming you’re not intending to mix up your attack moves) is one in (32 x 32 x 32 x 32 x 32)- or one pokemon out of over thirty-three million. If you want specifically physical attack or specifically special attack, that doubles to one in sixty-seven million, and if you’re looking for all-31s in the pokemon’s IVs, the chance is one in one billion, seventy-three million, seven hundred forty-one thousand, eight hundred and twenty four.
This chance is ridiculously small. You can alleviate it somewhat by breeding, but even with that, you can breed several boxes of a particular pokemon and only come up with a few pokemon that have 31s in two IVs, nevermind all six. Some people consider this worthwhile for competetive play, but even that tends to be pushing it. Why?
Well, the pokemon in question generally has an average of about two hundred and fifty points in a given stat. While the difference between an IV of 0 and an IV of 31 is a pretty significant amount (roughly twelve percent) of that, the difference between an IV of 28 and an IV of 31 is roughly one percent. At that point, you’re looking at a return on your time and effort so diminished that even in competitive play, the results are of questionable value. With a few exceptions for HP thanks to the way some effects divide hit points to determine their damage, and the major exception of speed (which determines who moves first and so is a stat in which a difference of 1 point can mean the difference between taking no damage and taking ½ your HP in damage), there is really not enough effect garnered from that one or two stat points to make the effort worthwhile.
It -is- true that you can luck out and have the one point mean the difference between a pokemon falling unconscious and that same pokemon getting a chance to recover or knocking its opponent out, but this happens rarely enough, especially at the level 100 area of play, that it’s not something you should reasonably be relying on. For some people, even these rare situations are worth the worry, and so the truth is that how much work you want to put in to finding the right IVs for your pokemon is entirely up to you, your luck, and your experiences as you play.
My personal guideline is that I breed roughly one full box of each pokemon that I intend to use for team members, and then whittle it down by comparing the stat values until I have three or four contenders- not always for the same kind of position on a team. This is how much time I personally am willing to put in in most situations. There are exceptions- if I have a particularly fast pokemon, like Crobat, that I want to make sure is at its top speed, I’ll run through two boxes for the best Speed stat I can get- but this applies only for Speed, because that one point of difference often means going first or not getting a chance to go. Even Health Points don’t matter on a one-point basis remotely that often.
Before deciding how far you’re going to go to get that perfect ability score, or how close to perfect you’re willing to accept, consider the environment you usually battle in. Battling against the computer is generally a foregone conclusion and you really aren’t going to need to invest any time in your IVs unless you’re unlucky enough to catch and try to use something that sports 0s or 1s in its defenses. Casually against your friends, the level of competitiveness varies and you’ll have to consider how close your fights tend to be. In competitive tournament play, however, you’re likely to need every significant edge you can get. Settle for a 30 or a 29, but don’t accept a 25 in that environment- there, the small differences can matter a lot.
