You have a Visa waver or a Visa, regardless they can refuse you to enter if they believe there is a 0.01 % chance that you will not leave the USA when your visa expires.
Visa does not allow you automatic entry to the US, border control can refuse if they believe you will not leave, you do not have the funds to stay, if they do not like your face, they can actually do what they want, if they believe there is a chance that you do not intend to leave the USA.