When T. H. Huxley coined the term 'agnostic' he based it on the Greek word gnosis (knowledge). He intended the term to mean a manner of examining knowledge, the practice of skepticism when determining whether or not to accept something presented as knowledge or 'truth' as valid and legitimate. Agnosticism deals with knowledge, knowing things.
Atheism is taken from the root 'theism', which is derived from the Greek (Theos) meaning “God”. Theism is the belief in a god or gods. Atheism is the rejection of or lack of a belief in gods. Both words relate specifically to belief in gods.
Gnosis/agnosticism...relating to knowledge and its acquisition. Theism/atheism...relating to belief, specifically belief in gods.
It's not only possible to be an agnostic atheist, I am one.
I'm agnostic (skeptical) in my approach to knowledge. I don't believe everything I'm told. I ask a lot of questions. I don't accept any knowledge as absolute.
I'm an atheist when it comes to believing in gods. I have experienced faith. I have believed in a god. I have listened to many people present the rational behind their belief and am no longer convinced. Theists insist their gods exist but can't provide irrefutable evidence to support their contentions beyond the assertion that it's a 'reality' only perceived by those who have faith and believe.
In other words they're saying that they can only prove what they believe is true to those who have already granted credibility to their claims. This is absurd. Nature has granted me the ability to think and question. Every time someone tells me that a thing is true, but tells me I can't understand this truth until I acquiesce to believing it's true, tells me that this truth cannot be questioned and if I'm presumptuous enough to go ahead and question it anyway (blasphemer!) I'll only encounter riddles, metaphors and enigmatic, fantastic stories, tells me that only the 'enlightened' can grasp this truth but that to become enlightened I must join this group or perform that ritual, buy this book or attend that lecture...it's been a scam. All the religious truly have to offer is a like-minded and often close-knit community, rituals (and humans do enjoy rituals. The most secular society will still have state rituals. Even individuals have their rituals) and the peace of mind that comes from abdicating all responsibility for your life.
I prefer a community I choose for myself, I prefer my own rituals and am willing to accept responsibility for my own life, for the decisions I've made (bad and good) and what's become of it. I have no need for what religion offers. I don't want to waste my money on buying what faith is selling.
I'm an atheist because I used agnosticism to examine the claims of religious belief and found them unconvincing and invalid. I'm an agnostic atheist.