There are several instances, it just depends on the context of your pokemon, but here are a few instances which can prove useful.
-If you want a priority move in addition to a reliable STAB, there are times it can be good to carry two moves. An example would be Infernape carrying Mach Punch and Close Combat. Mach punch is good should he need to take something out quickly, but Close Combat is there for the stronger pokemon he doesn't have to worry about outspeeding.
-If you have an ability that compliments the situation. For instance, carrying Eruption and Fire Blast on Typhlosion (with the ability blaze) can be nice. When his HP is high, he can use eruption, then switch to fire blast when his health falls, activating blaze.
There are tons of other instances, it's just a matter of what the moves provide besides the STAB. If they both just offer STAB and type coverage, then you don't need two moves. If they offer some other benefit, such as priority, then that would be a good instance to use more than one STAB move.