Why are hybrids difficult to design?

At the heart of the hybrid problem is the fact that if a hybrid can perform a given role as well as a specialist while also having other abilities the specialist can never have, playing a specialist becomes pointless.
To put it in terms of our earlier example, if a paladin can tank as well as a knight but can also heal, then there is never a reason to play a knight instead of a paladin. If the hybrid has all of the advantages of its parents plus extras, then the parent class is doomed to extinction.
Conversely, if a hybrid is always inferior to a specialist in any given role, then it's always better to have a specialist fill that role. As game designers, we want to create a vibrant ecology of classes, where players have a wide variety of classes and playstyles available to them.
The standard solution to this problem can be summed up in the phrase "Jack of all trades, master of none". Hybrids are generally made less effective in each area than their parent classes, with the intent that they make up the deficiency with their abilities from other areas. The paladin mentioned above might not be able to survive as much damage as a knight, but they can heal other players and help them survive, something a knight could never do.
Historically, MMOs have had a great deal of difficulty designing hybrids that are powerful and valuable without completely displacing their parent classes. The catchphrase for these overly successful hybrids is "tank-mage". This term comes from the early days of one of the first MMOs, Ultima Online, where some characters could both wear heavy armor and cast powerful damaging spells. A tank-mage could both take and deal a lot of damage, creating a character that was superior to any other type of character in most situations.
Since Ultima Online, other MMOs have tried to avoid this problem, but players inevitably gravitate towards the latest incarnation of the tank-mage whenever possible. This is not a sign that the players are cheating or deliberately trying to abuse the system, it's just the natural result of players trying to find the golden path and "win" the game. A character who can take more damage is better and a character who can dish out more damage is better, therefore a character that can do both is ideal.
For example, in City of Heroes the Fire/Fire Tanker emerged as an early tank-mage contender because of its high damage resistance and ability to deal lots of damage to multiple enemies at one time.
The discovery/creation of these tank-mages by players is the product of the incredible ingenuity of MMO players and the complex emergent properties of the game systems, rather than any particular failure on the part of the design teams, and they are mostly adjusted or "nerfed" as they become apparent. However, nerfs generate turbulence in the player community, and it is always better to prevent these sorts of issues from arising in the first place.
The hybrid issue is exacerbated by the fact that MMOs are both solo and group games. If people only played MMOs in groups, a character able to soak a lot of damage but deal no damage would be viable because the other people in the group could deal damage for them. The individual character could be one dimensional (a pure tank) because the other group members fill out the other two parts of the trinity (DPS and healing).
However, studies have shown that even in group-focused games, players spend a lot of their playing time doing things on their own. Even if a character is the best healer in the world, if they can't take or dish out at least some damage they won't be able to operate outside of a group. Soloing requires that the character be able to deal damage, plus the ability to absorb, avoid, or heal the damage taken.
Therefore, once the design decision has been made that every character should be able to solo, a decision that has been made practically mandatory by the successful example of World of Warcraft, it automatically follows that every character must be a hybrid and therefore subject to the paradoxes of hybrid design. This is a universal problem, not just one that affects certain classes within a game.