They will certainly look more closely at your second application to verify the genuine nature of your relationship and confirm that there is no marriage fraud. It is ok for a person to get married more than once in their lifetime; it is not ok to marry for the wrong reasons, so the job of the immigration is to make sure that this is a legitimate marriage.
Answer Applies to: Washington
Read the instructions to the forms carefully. They address this issue.
Answer Applies to: Florida
If your marriage is bona fide, yes.
Answer Applies to: Nevada
If your past and current marriages were entered into in good faith you should have no issue. You may be required to submit evidence to show both marriages were entered into in good faith.
Answer Applies to: New York
You will be examined very closely by INS on any subsequent K-1 visa after the first one. It's likely that your first marriage and divorce will be checked out thoroughly, including the grounds for divorce, the timeline of when your first spouse came to the country and when the separation and/or divorce occurred, in order for marriage fraud to be ruled out. There are provisions in the law that require you to ask for a specific waiver for marriages of this type after the first one. You should seek the advice of a good immigration lawyer to help you get through this minefield, as the consequences of a finding of marriage fraud can have serious immigration consequences.
Answer Applies to: Iowa