It depends on the group and expectations about this kind of stuff before play begins...as many others have pointed out. Usually, you want to have a sense of these things before you start to play so that if anything comes up, everyone has a good idea how it will be handled.
It also depends on the game. This is a thread in the D&D forum, but even with D&D you may have different ways that the DM will run things, different approaches to play, that will dictate this kind of thing.
Because some versions of D&D (I'm using "versions" here as shorthand for both edition and GM/table specific practice) have strong niche protection. And some versions of D&D are about a set scenario (a module or adventure path), so the obstacles are predetermined with no thought to the actual characters that may wind up in play. This results in a broad range of obstacles, which will require a broad range of PCs due to niche protection. So with this kind of approach, it's best to create a team, with well rounded skills spread out across all characters, hoping to address as many specialties as possible.
With a version of D&D that's not about a set scenario, but is instead tailored for a specific group of characters, that team element may not be as prevalent. In this kind of approach, players may be more free to simply choose who they want to play, and not concern themselves with "filling the gaps".
And this all applies to character goals and the like, too. The more team focused the game is, the harder it is for characters to pursue their individuality. It's certainly not impossible, it's just trickier given the nature of the game and the party or team element.
Other games...either highly modified versions of D&D or else other games entirely...the game maybe all about individual goals. Some games actively have the players competing with one another. These games can work totally fine, largely because they largely eschew the team mentality.
There are a lot of possibilities. My personal approach will vary from game to game. When it comes to D&D, I tend to approach it with the tailored approach, so that the characters are free to have a good deal of individuality, but they can still function as a team. The game is just kind of set up that way, so if I'm going to shift far away from the party or team approach, then I'm simply not going to use D&D.