Immunity to one part of a power does not make a creature immune to other parts of the power. For example, when a creature that is immune to thunder is hit by a power that deals both thunder damage and pushes the target, the creature takes no damage, but the power can still push it.
Against Combined Damaged Types A creature's resistance is ineffective against combined damage types unless the creature has resistance to each of the damage types, and then only the weakest of the resistance applies.
Arlough, the online Compendium is incomplete in the rules text of the Immune entry from both the PHB3 and Rules Compendium.
Immunity to one part of a power does not make a creature immune to other parts of the power. For example, when a creature that is immune to thunder is hit by a power that deals both thunder damage and pushes the target, the creature takes no damage, but the power can still push it.
So, if you are immune to one damage type and you get hit by a power that has two damage types, you are not immune to any of the damage. If a creature is hit with a power that deals 20 Cold/Necrotic damage and it has Immune Cold and Resist Necrotic 5, it will take 15 damage, because it is not immune to all parts of the power and has to use its lowest resistance against the power (since immunity is really just a level of resistance).

I believe in the very last sentence the rule above actually states specifically that, in the case of a power of multiple types, one of which you have an immunity to, you will not deal any damage, but you will be able to execute effects of that attack. So in the example of the power that deals 20 Necrotic/Cold and the creature has Immune Cold and Resist Necrotic 5, it will take no damage at all.

(Dungeons & Dragons)
Rulebook featuring "high magic" options, including a host of new spells.