This is defined in the settings screen for each Scheduler. You can choose from either "Students can make 1 appointment at a time" or "Students can make 1 appointment in this scheduler" (or, in version 2.7 and beyond, you can also choose "2 appointments at a time" etc.).
If you have chosen "Students can make 1 appointment in this scheduler", then exactly this should happen - once a student has booked a slot, they can not book another one. The can change their previous booking, though.
If with these settings it doesn't work for you, then please let me know about details of your configuration (which version of Scheduler are you using, what settings did you choose, what does the student see - screenshots would be helful).