Nope. You would get $10,000 when you get to your first duty station (and the paperwork gets processed)...then the rest is divided so that you get part of it (minus tax) on every anniversary of your enlistment.
This is not how mine was broken down. I received a 20k bonus for a 3 year enlistment. It was split up into 3 installments. 1st one was given to me while I was towards the end of my AIT (7,000), 2nd was a year later (7,000), and the third was given to me the following year (6,000). A couple of the guys I went to AIT with were slightly different based off of their enlistment length. I think one of them actually received it the way you are mentioning because I heard him complaining he didn't get as much the 2nd go around.