If my bill is, for example, $9.63 and I only have a $20 on me, I expect to get $11.37 in change. I will then leave a $2 tip, which is what I leave for checks in the $10 range - I tip a little higher on small checks because the server has had to do the same amount of work. It isn't their fault that I ordered something cheap.
If I don't get the coins back, I'm not leaving a further $2 as tip. At most, I'm leaving an additional dollar. So a server not bringing me all my change? Is going to get less of a tip.
It really isn't as big a problem here in Canada, though, because our smallest bill is a $5. So servers have to bring coinage, regardless, and so bring all the change. And if the change is $15.37, you'll quite often get a $10, 1 or 2 $2 coins and 3 or 1 $1 coins as your change so you can leave a tip. If they bring you a $10 and a $5 bill, you might not have the small stuff to leave a tip.