...which stands for Health Reimbursement Arrangement or an FSA (flexible spending account) and still be HSA eligible.  There are some rules around this which we are not going to cover here.

Say your back goes all wiggly and you need to go see your favorite back snapper, Dr. Flickfinger.  Unfortunately, your high deductible health plan (HDHP) does not cover chiropractic, but lucky for you, your employer contributes to an HRA (or FSA) on your behalf.  Say you rack up (bad choice of words?) $2,000 in back cracking that you are able to pay for out of your HRA.

Do the dollars you spent getting "the treatment" count toward satisfying the deductible on your HDHP?

Nope.  The rule is that only medical expenses covered by the HDHP can count against the deductible. 

If you have funded your HSA, it is worth noting that it may be more flexible than your insurance policy in terms of what it will pay for.  You have more control over it, plus you get the tax and interest earnings benefits it offers.  For a more comprehensive list of things you can spend your health savings account dollars on there is a pretty good laundry list located on the "Resources" page of HSAeducator.com.