Take a look at these 6 Pokemon and their weaknesses:
Whiscash (Grass)
Scizor (Fire)
Spiritomb (Fairy)
Clefable (Poison or Steel)
Chansey (Fighting)
Misdreavus (Ghost or Dark)
Any combination of 5 moves would not be able to hit all 6 of these Pokemon super-effectively. Instead, 6 moves would be needed, either:
Grass, Fire, Fairy, Poison, Fighting and Ghost,
Grass, Fire, Fairy, Steel, Fighting and Dark,
Grass, Fire, Fairy, Steel, Fighting and Ghost, or
Grass, Fire, Fairy, Poison, Fighting and Dark.
Therefore, it is not possible to hit every Pokemon super-effectively with just 5 moves.