Negative. On third down the job of whoever gets the ball is to move the chains and maintain possession. 3rd and short is the great RB down, they hit the line keep the legs moving and get it done. 1st and 2nd downs are for homeruns sometimes (nothing wrong with plodding clock eating drives, the Pats have gotten 3 rings in a large part thanks to those), 3rd down is for getting the team another 1st down.
Ok Coach...
If it's third and three or less, there isn't a running back in NFL history who I'd rather have in my backfield than the Earl of Texas (Earl Campbell).