Sure, just like you can't charge extra for using a credit card in California as payment in most scenarios (you can get a cash discount though). There may not be late fees, but the hospital will absolutely offer to settle for a hefty discount if you pay ASAP.
By the time you get something claiming "PAST DUE" the hospital may be less willing to negotiate.
Who said anything about getting nothing? If the hospital sells your debt to a collection agency they get something. If the hospital deals with it in house they can act like UVA[1] and sue you to garnish your wages or take out a lien on your property. Playing chicken with the hospital billing department is a bad way to deal with medical debt.
1: https://www.washingtonpost.com/health/uva-has-ruined-us-heal...
If you really did just get the bill it will be several months (and several more copies of the bill sent to you) before a final bill would show up that would indicate the account is "PAST DUE" and heading for collections.
IANAL but you can safely wait for the itemized bill.
In the future to save on costs, you might try to find a 24-hour Urgent Care instead of full-fledged ER. Unfortunately the level of equipment that they have to keep in a full ER means just having someone in a bed costs them a lot of money, even if they are just swabbing your throat for a strep test.