In general Pokémon can only learn four moves at a time. Pokémon Mystery Dungeon keeps with this tradition, but Pokémon can learn the same move multiple times.
The immediate advantages include greater endurance and longevity for a move before the consumption of Max Elixir. The duplicate moves, however, are otherwise treated as separate moves (and thus occupy one of the four slots) rather than one single move with greater PP. Having a duplicate of a move also allows for a higher rate of usage of that particular attack, regardless of PP. Other aspects of a duplicate move occupying a separate slot involve the isolation from dungeon traps, which might disable a move, and powering a move up with Ginseng.
This feature is immediately granted and easily exploited with TM usage. For example, the use of Shock Wave twice on a Pokémon allows the Pokémon to use more than its original 8 PP of a move that has 100% accuracy.
There are also wild Pokémon who have duplicate moves due to the move appearing twice in the Pokémon's level-up movepool. One example includes a few Pupitar in the Southern Cavern with two Sandstorm moves.
Effects of evolution
It is also possible to raise a Pokémon to learn two of the same moves that aren't in TM form. Since different evolutions of a single Pokémon learn moves at different levels, the method through which Pokémon evolve in Mystery Dungeon allows for natural duplication. Pokémon will only evolve after being brought to Luminous Cave while satisfying their normal evolution criteria. If the Pokémon has learned a move its evolution would not have learned at the current level, after evolving, it will learn the desired move again as it levels up.
For example, a Squirtle can be raised until it learns Hydro Pump. After learning the move, it can be evolved into a Wartortle. Wartortle can then be raised until it learns a second Hydro Pump. Then, it can be evolved a final time and leveled up for a third Hydro Pump. This process results in a Blastoise with Hydro Pump in three move slots. Another example is a Metagross, which can undergo a similar process to obtain two Meteor Mash attacks, or two Agility moves.
Moves cannot be duplicated while in the Gulpin Link Shop. Once a non-TM move has been duplicated and deleted it cannot be learned again for the purpose of duplication unless the Pokémon naturally learns it again at higher level. It is also possible to link two similar moves for redundant effects.
In Pokémon Mystery Dungeon: Explorers of Time and Explorers of Darkness, the ability to learn duplicate moves was removed.