In terms of scaling, there is a natural limit to how the base game can play out with 2, 3, or 4 players. Moving beyond that natural limit requires changes to the board for the game to remain playable. The board simply becomes too small for a 5th or a 6th player unless the board is expanded. Also, extra rules involving when players can build have to be implemented for the system to not break down under the weight of a 5th or 6th player.
This example of Settlers illustrates an important point: most game systems have natural limits with respect to their player capacities and most systems need to be altered in some way for them to expand upwards or downwards from their natural limits with respect to the number of players they can accomodate. It's possible that many games put out on the market that are 2 to 4 player games could be made into 2 to 6 player games but the alterations to the game's basic system are such that offering a base game with that level of scalability is simply not financially a smart move. The publishers would have to put a higher price on the game because of the extra components needed to allow the game to scale.
In terms of a game's upper range with respect to scaling, some game systems simply become too "crowded" if you add too many players - thus the player range remains restricted. Let's take Saint Petersburg for example:
In this game, there are four distinct phases: workers, buildings, aristocrats, and upgrades. There are also 8 slots for cards on the board. With Saint Petersburg's game system as it stands, it comfortably plays anywhere from 2 to 4 players. However, if you were to try to include a 5th player, the problem of starting first in a phase each round comes into play. With only 4 phases and 5 players, one player does not get to start first in a phase which could be considered unfair. Also, with a 5th player, the board would have to be made larger to allow more slots for cards. Further, the number of cards in each deck would have to change as the depletion of any one deck is a trigger for the end of the game. Thus, the game system as it stands has a natural limit of 4 players and, to go beyond that, would require some significant changes to the game's basic system.
Triangulation and Non-Triangulation Mechanics
Now let's look at the other side of scaling. Some game systems have a natural limit on the lower end - meaning the game breaks down unless there is at least a minimum number of players. If a game has to have at least 3 players for it to be realistically playable, then it's probably due to a "triangulation" mechanic forming a key part of the game's basic system. Graphically, here is the difference between a triangulation relationship and a duelistic one:
What becomes immediately apparent is that adding one additional player greatly increases the interactive complexity of the system. Such interactive complexity is necessary for certain game systems to function. Let's look at two specific examples of this:
Many Eurogames implement some kind of auction. Most games that incorporate an auction as a fundamental part of that game's system will have, as a requirement, at least 3 players. There are exceptions (Power Grid is the first to come to mind) but, for the most part, auctions are only realistically interesting if there are three people involved.
Large Payout due to Attrition
Certain games offer a large payout if a player does not succomb to attrition. Examples of this include such games as Diamant/Incan Gold and Cloud 9. You will also notice with these games that a minimum of 3 players is the case. Again, the reasons for this are obvious. If there are only two players, the opportunities for big payoffs are too easy to come by and the competitive tension of the game is tremendously crippled.
Some game systems require a specific number of sets of resources in play in order for the game to work. I'm specifically thinking of Blokus in this respect. To anyone who's played the game, it makes sense that four sets of pieces must be in play for the game to have the tension that it does. Otherwise, if there were fewer players, then either each player would have to have extra pieces that they wouldn't use in a game where there are a larger number of players, or the board would have to be reduced in size (which explains why the game Travel Blokus was published). So, those four sets of pieces remain in play but the rules about who controls which set at which time allow the game to scale from 2 to 4 players.
Most common examples of "locked" systems in games are of the two player variety: Chess, Yinsh, Go, Lost Cities, etc.
Large Group Games or "Party" Games
Some game systems depend on a much larger number of minimum players to work. Werewolves, for example, really needs at least 8 players to have the proper amount of tension it needs. Most party games in general usually need a larger number of players to really work. Otherwise, with only two or three players in a typical party game, the socially interactive element is not robust enough to carry the time and, thus, more pressure is put on the game's intellectual intrigue to provide fulfillment. This can be a problem as most party games are built around the concept of achieving fun via amusement rather than through intellectual intrigue.
Further, many party games that are built on a duelistic system incorporate the idea of competing teams - which implies a minimum of at least 4 players to make the game viable. Other games rely on a triangulation system to work. Apples to Apples is a prime example of this (imagine playing it with only two players).
Impact on Game Design
Regardless of what kind of game system a designer has in place, it's important for the designer to recognize which aspects of his or her prototype need to adjust in the scaling process and which aspects need to remain the same. It's easier to identify those aspects if the designer is aware of the inner dynamics at play within a game system so as to preserve the tension of the game as each new player is added to the equation.