It depends on your definition of consciousness. If we use the definition on Wiki (which I think is a good one), I'd argue that 'processing power' (of the brain) plays a role in our consciousness experience.
From Wiki: "
Consciousness, at its simplest, is
sentience and
awareness of internal and external existence"
'Awareness of internal existence', depends on how the brain processes data. Our thoughts (awareness of internal existence) are dependant upon how smart/intellectual we are, which is directedly related to processing power. To quote Socrates: “Strong minds discuss ideas, average minds discuss events, weak minds discuss people.” I'd assume that most, if not all of those who frequent these types of forums belong in the first category. A 'strong mind' is one that has more processing power than those with 'weak minds', and it's the processing power that determines how deeply we can understand a subject, which is no doubt related to our interest in it (intellect). It's also why other organisms, such as dolphins can only reach the intelligence of an average 5 year old, since they don't have the 'processing power' to go beyond that level of intelligence.
Obviously, processing power isn't the totality of one's conscious experience, however it's definitely related to it.