Patients with amnesia usually know who they are, but they have problems storing new memories. You wouldn't realise this from the movies. Films like The Bourne Identity show us the opposite pattern - characters who have forgotten who they are, but who have no trouble with their everyday memory.
Attention is, by definition, limited. And that's usually a good thing. If you're searching for a lost earring on the floor, you want to ignore anything that's not small and shiny. When talking to someone at a party, your mind helpfully tunes out all of the other voices prattling on around you.