No. I used to carry mail, and in some places the boxes are picked up very early (I had one picked up at 1030 am) based upon the carrier's route.
The post offices themselves vary based upon the contractor route (that is, the company responsible for picking up the mail at that office and taking it to the central office). This may be earlier still, especially for small towns that are a way from the main office. For example, Springfield, Missouri handles the mail from the Oklahoma line all the way to within 100 miles of St. Louis. Bourbon, MO (the last town handled by Springfield NE on I-44) goes all the way to Springfield to be postmarked and sorted. I don't know the pickup time for them, but I'm sure it's pretty early in order to get it 150 miles away to make the connections to get the mail where it is going.
You can go to www.usps.com and click "locate a post office". I believe (but don't promise) that it has a schedule on it. You can also call them and they should be able to tell you.