If the ask is at $8.03 and you are first in queue at $8.01 all the orders in $8.03 and $8.02 have to be filled before you get filled so you have plenty of time to cancel your order.
Assuming you mean bid. If the the size is only 10 @ 8.03 and 1 @ 8.02 and 8.01, then if someone is looking to sell 100 at market price it will be a problem. The sell order will go through many levels after filling yours and it will all happen at the same time.