Please help us help you and fill where relevant:
Your LimeSurvey version: 3.28.42
Own server or LimeSurvey hosting: LimeSurvey hosting
Survey theme/template:
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Hello, I am trying to help our LimeSurvey users determine if they can use the product for registering students for training courses. So setting up a survey with questions for them to fill out like Name, Training Class, Date, etc. They are asking if it's possible to limit the amount of people who register. For example, if a registration survey is created and sent to 150 people but the class only holds 80. Can the survey not be taken once it hits 80 responses (an error shown to the 81st attempt). Is there then a way to see who attempts to take the survey after that (so they can create a wait list)?  
Thank you in advance for any insight you might have to this type of functionality!

 
Please read the manual.

Quotas - www.limesurvey.org/manual/Survey_quotas/en

As Tpartner says, with quotas you can limit the number of people and they will be redirected to a individual page for the quota, where you could add the error message.

To create a waiting list you could create a second survey. You could put the link to this waiting list survey on the page of the quota.

However, keep in mind that a quota will only close once there are for example 80 complete registrations. Now if the form is fairly long and a lot of students try to register at the same time, it could happen that student number 80 is still filling the form and student 81 is already filling the form. Quota can not solve this problem. Student 81 will still be able to fill the form, because the quota isn't full at the time student 81 enters the form.

So to avoid this, you probalby want to set the number a bit lower and fill up with students from the waiting list later.