No, DR and resistances do not stack.

DR only protects against the physical damage type. Resistances apply to other damages types, like fire, acid, electricity and so on. There is no overlap between damage types.

In your example, with 10 physical + 6 electricity damage:

• A monster with DR 5/- would receive 5 physical and 6 electricity damage.
• A monster with DR 20/- and electricity resistance 5 would receive 0 physical and 1 electricity damage.

The reference you're looking for is on page 197 of the Player's Handbook, under, naturally enough, Damage Resistance and Vulnerability. Specifically the third paragraph states:

Multiple instances of resistance or vulnerability that affect the same damage type count as only one instance.

So the Bear Totem Barbarian still takes half damage from bludgeoning, slashing, or piercing attacks.