Is it possible for a person to have anterograde amnesia and be aware of it? In "Memento" the main character has got anterograde amnesia, but he's aware of it. Is it even possible and how, if it is? Compare it to "50 First Dates" &mdash; the heroine is not aware of her condition.

Imagine you wake up in the morning. How would you come to awareness that you've got anterograde amnesia, if you don't remember anything since the moment you got it, including that? And even if you find this out by logical thinking, you'll forget it soon (in "Memento", what, in several minutes?)
Yes it is possible. It is very common for normal individuals to feel like they should remember something, but are then unable to actually recall the contents of that particular memory. This feeling of knowing is called metamemory.

This study by Art Shimamura and Larry Squire found that 


  patients with severe anterograde amnesia can produce accurate
  feeling-of-knowing judgments. Thus, impaired feeling-of-knowing
  accuracy is not an obligatory feature of amnesia


So, some of those patients may wake up each morning and soon realize that something is amiss, as they are unable to recall facts or events that they feel they should be able to recall.