Hello Lenny,
We get the same (or a similar) issue all the time when we're fully booked.
The reason customers see a message that says you have no availability past a specific date can be due to the limitations on how far out someone can book.
If you go into your schedule in Acuity, then click on Calendar Scheduling Limits, look for Do Not Allow Clients to Schedule Few Than [ box to put a number in] hours in advance and more than [box to limit how far out they can book] days in the future.