2000' 'high' at FL340 is nothing. Today, descending through about FL230-240, we were 2400'+ above path with a 250/12000 restriction ahead of us.
FLCH(Airbus open descent??), idle, increased IAS from 280 to 320-325, and the increased drag from 320-325 vs. 280 KIAS used up the 2400'+ of vertical track error(above VNAV PATH).
If you were 2000' high at FL340 and the Captain demanded full speedbrakes he's pretty insecure about his ability to manage the airplane's energy.