They will usually make your visa valid from the date you put on the application, so if they make it valid from 16th June, then no, you can't travel before this date (but you can travel any time after that date). If they happen to make the visa valid from the date they issue it, and it's before 16th June, then you can enter on that date instead.
Have you actually sent off the visa application to NYC yet? If not, you can manually cross out the 16th June date on your printed application form and change it to a different date. No one will see the application until they receive the physical copy that you mail to them.