I'm actually not sure if you can have multiple classes, but I know you can make your own class.
Actually there are 4 set classes; Soldier, Operative, Medic, and Engineer. You can't make up a new one. These classes are the ones that will be final (so far anyways). You can however
customize how they function via the Abilities (or Perks?) loadout screen, in the main menu. Each ability allows a bit more diversity in your class. But your original statement is wrong. You cannot make a new class. No Medics disguising as the enemy, blowing up new passage ways, with your turret. Sorry to let you down.